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ELECTRIC   AND   MAGNETIC 
MEASUREMENTS 


THE  MACMILLAN  COMPANY 

NEW  YORK  •    BOSTON  •    CHICAGO  •    DALLAS 
ATLANTA   •    SAN   FRANCISCO 

MACMILLAN  &  CO.,  LIMITED 

LONDON  •    BOMBAY  •    CALCUTTA 
MELBOURNE 

THE  MACMILLAN  CO.  OF  CANADA,  LTD. 

TORONTO 


ELECTRIC  AND  MAGNETIC 
MEASUREMENTS 


BY 


CHARLES    MARQUIS   SMITH 

ASSOCIATE    PROFESSOR    OF    PHYSICS 
PURDUE    UNIVERSITY 


gorfc 

THE   MACMILLAN  COMPANY 
1917 

All  rights  reserved 


COPYRIGHT,  1917, 
BY  THE  MACMILLAN  COMPANY. 

Set  up  and  electrotyped.     Published  February,  1917. 


Kortoooti 

J.  8.  Gushing  Co.  —  Berwick  &  Smith  Co. 
Norwood,  Mass.,  U.S.A. 


PREFACE 

THE  present  work  is  the  development  of  a  course  of  lectures 
and  laboratory  notes,  which  have  been  used  for  some  years  by 
students  in  mimeographed  form.  The  equivalent  of  one  year 
of  general  physics  and  some  knowledge  of  the  calculus  are 
presupposed ;  hence  the  arrangement  and  treatment  are  not 
always  that  which  would  be  followed  with  a  class  of  beginners. 
Much  of  the  material  has  been  arranged  from  lecture  notes, 
and  the  standard  textbooks  and  treatises  have  been  freely 
drawn  upon.  Most  of  the  laboratory  exercises  are  described 
in  such  a  way  that  particular  types  of  apparatus  are  not  de- 
manded, unless  these  are  well  known  and  generally  available. 

The  methods  described  are  for  the  most  part  standard  ones, 
with  such  variations  as  many  years  of  trial,  and  suggestions 
from  various  instructors  associated  with  the  course  have  shown 
to  be  of  value.  To  no  one  individual  can  the  credit  be  ascribed 
for  the  present  form  of  the  book,  as  it  has  had  the  advan- 
tageous criticism  of  numerous  instructors  of  varied  degrees 
of  experience. 

In  some  instances,  quantities  have  been  mentioned  before 
they  have  been  formally  defined.  This  practice  seems  defensible 
from  two  standpoints.  Usually,  the  term  employed  is  one 
which  the  student  may  be  expected  to  recognize  from  his  pre- 
vious work  in  general  physics.  Again,  it  is  sometimes  well  to 
direct  the  attention  of  the  student  to  a  term  or  an  expression 
before  it  is  defined,  so  that  the  definition,  when  it  is  given,  will 
satisfy  an  aroused  interest. 

The  articles  on  dimensional  equations  should  be  made  use 
of  as  occasion  may  arise,  and  in  any  event  not  until  the  appro- 
priate chapters  have  been  reached. 

v 

393938 


vi  PREFACE 

In  general  the  student  should  be  required,  in  every  labora- 
tory exercise,  to  state  the  percentage  of  accuracy  of  his  read- 
ings, and  to  note  the  influence  of  the  errors,  singly  or  combined, 
in  the  final  result. 

The  author's  thanks  are  due  to  Mr.  G.  W.  Sherman,  who 
has  made  the  drawings  for  the  book  and  who  has  read  the 
entire  manuscript ;  also  to  Mr.  A.  F.  Wagner  and  Mr.  C.  G. 
Watson  for  many  helpful  suggestions  throughout  the  prepara- 
tion of  the  book,  and  to  the  Leeds  and  Northrup  Company  for 
permission  to  use  cuts  of  their  apparatus. 

C.  M.   SMITH. 


CONTENTS 


Introduction 

Part  I.  Laboratory  Methods 

Part  II.  Fundamental  Definitions  and  Units 

Part  III.  Dimensions  and  Dimensional  Formulas 

Part  IV.  Miscellaneous  Information      .... 

Chapter  I.    Galvanometers    .        .       .       . 

Part  I.  General  Types  and  Characteristics 
Part  II.  Descriptions  of  Special  Types  of  Galvanome- 
ters   

Part  III.  Shunt  Circuits 

Part  IV.  Sensibility  of  the  Current  Galvanometer 

Part  V.  Potential  Galvanometers  and  Voltmeters 

Chapter  II.    Resistance  and  Its  Measurement     . 

Part  I.  General  Definitions 

Part  II.  Measurement  of  Resistances  of  Medium  Value 

Part  III.  Measurement  of  Low  Resistances  . 

Part  IV.  Measurement  of  High  Resistances 

Part  V.  Measurement  of  Liquid  Resistance 

Chapter  III.  Electromotive  Force  and  Potential  Difference 

Part  I.  Sources  of  Electromotive  Force 

Part  II.  Potentiometers 


Chapter  IV.  Electric  Current 

Chapter  V.    Capacity  and  the  Condenser      .        . 

Part  I.  Definitions  and  Units 

Part  II.  Charge,  Current,  and  Energy  Relations 

Chapter  VI.  Electromagnetic  Induction 

Part  I.  Mutual  and  Self  Inductance   . 

Part  II.  Current,  Energy,  and  Charge  Relations 


1-17 

1-4 

5-9 

10-15 

16-17 

18-52 
18-26 

27-31 
32-40 
41-44 
45-52 

53-100 

53-66 
67-78 
79-88 
89-95 
96-100 

101-138 
101-122 
123-138 

139-160 
161-177 
161-170 
171-177 

178-203 

178-189 
190-203 


Vlll 


CONTENTS 


Chapter  VII.    Electrical  Quantity  and  the  Ballistic  Gal- 


vanometer        

204-220 

Chapter  VIII.    The  Measurement  of  Capacity      . 

221-237 

Chapter  IX.    The  Measurement  of  Self  and  Mutual  Induc-^ 
tance         

238-263 

Part  I.       Self  Inductance        ...... 
Part  II.     Mutual  Inductance           ..... 

238-254 
255-263 

Chapter  X.    Magnetism  and  the  Magnetic  Circuit 

264-283 

Chapter  XI.    The  Earth's  Magnetism  .        .        .        . 

284-300 

Chapter  XII.    Magnetic  Testing  

Parti.       Magnetization  Curves  —  Hysteresis 
Part  II.     Methods  of  Magnetic  Testing         .         .    '     . 

301-360 

301-319 
320-360 

Appendix        
Part  I.       Absolute  Measurements           .... 
Part  II.     Tables      
Part  III.    Standard  Reference  Books       

Index 

361-367 

361-365 
366 
367 

369-373 

LIST   OF   LABORATORY   EXERCISES 


SEC.  PAGE 

I.     A  study  of  galvanometer  types  ....       25          30 

II.  To  study  the  effect  of  shunts  on  galvanometer  de- 
flections   28  40 

III.  To  study  the  sensibility  of  a  current  galvanometer    .31          43 

IV.  To  calibrate  a  galvanometer 32          44 

V.     Some  experiments  on  Ohm's  law        ....       36  50 

VI.     To  measure  a  resistance  wiih  the  meter  bridge  .61          71 

VII.     To  measure  a  resistance  with  the  box  bridge  53          74 

VIII.     To  verify  the  laws  of  series  and  parallel  resistance 

combinations .54          75 

IX.  To  measure  a  low  resistance  with  the  Kelvin  bridge, 
and  to  find  the  temperature  coefficient  of  resist- 
ance for  a  sample  of  wire  62  86 

X.  To  measure  a  low  resistance  by  the  fall  of  potential, 
direct  deflection  method,  and  to  find  the  resistiv- 
ity of  a  sample  of  wire 63  86 

XI.     To  measure  a  low  resistance  with  ammeter  and  milli- 

voltmeter 65          88 

XII.     To  measure  a  high  resistance  by  the  substitution 

method 70          94 

XIII.  To  find  the  conductivity  of  an  electrolyte  .         .         .       73          98 

XIV.  To   calibrate  a  galvanometer  used    with   a  copper- 

advance  thermo-element      .....       82         110 
ix 


LIST  OF  LABORATORY  EXERCISES 


XV.  To  study  the  variation  in  the  potential  difference 
at  the  battery  terminals,  and  to  find  the  in- 
ternal resistance  of  the  battery  cell  by  the 
galvanometer  method  .....  85 

XVI.     To  measure  the  internal  resistance  of  a  battery 

by  the  voltmeter-ammeter  method          .         .       86 

XVII.     To  compare  electromotive  forces  by  the  condenser 

method 87 

XVIII.     To  measure  the  internal  resistance  of  a  battery 
by  the  condenser  method        .... 

XIX.     To  make  a  time  test  of  a  battery  cell    . 

XX.  To  measure  electromotive  force  with  the  simple 
potentiometer  and  with  the  resistance-box 
potentiometer 

XXI.     To  compare  electromotive  forces  with  the  Wolff 
potentiometer 

XXII.     To  compare  electromotive  forces  with  the  type  K 
potentiometer          ...... 

XXIII.  To  measure  current  strength  with  the  potenti- 

ometer, and  to  calibrate  an  ammeter 

XXIV.  To  measure  a  high  voltage  with  the  potentiometer, 

and  to  calibrate  a  voltmeter  .... 

XXV.     To  determine  the  mechanical  equivalent  of  heat 
with  the  flow  calorimeter       .... 

XXVI.     To  determine  the  constant  of  an  electrodyna- 
mometer  with  the  copper  voltmeter 

XXVII.     Calibration  of  a  moving  coil  ballistic  galvanometer 
with  a  standard  condenser      .... 

XXVIII.     Calibration  of  a  ballistic  galvanometer  with  a 
standard  mutual  inductance  .... 

XXIX.     To  determine  the  constants  of  a  magneto-inductor    153 

XXX.     To  compare  capacities  by  the  direct  deflection 
method  ........ 

XXXI.     To  study  leakage,  absorption,  and  residual  charge    157 


116 
117 

118 


88 

119 

90 

120 

93 

126 

96 

130 

97 

134 

98 

134 

100 

137 

112 

156 

114 

159 

150 

215 

151 

217 

153 

219 

156 

224 

157 

225 

LIST  OF  LABORATORY  EXERCISES  xi 

SEC.  PAGK 

XXXII.     To  compare  capacities  with   the   Wheatstone 

bridge.    First  method          .        .         .         .159        229 

XXXIII.  To   compare  capacities  with   the    Wheatstone 

bridge.     Second  method     .         .         .        .160        280 

XXXIV.  To  compare   capacities  with   the   Wheatstone 

bridge  and  vibration  galvanometer     .         .     161        231 

XXXV.     To  compare  capacities  by  the  method  of  mix- 
tures         163        234 

XXXVI.     To  measure  a  high  resistance  by  the  loss  of 

charge  method 165        237 

XXXVII.     To   determine  an   inductance   in  terms  of   a 

capacity 168        243 

XXXVIII.     To  determine  an  inductance  by  a  modification 

of  Anderson's  method          ....     170        247 

XXXIX.  To  measure  a  self-inductance  by  comparison 
with  a  variable  standard.  Vibration  gal- 
vanometer method 173  252 

XL.     A  study  of  mutual  inductance  .         .         .         .175        255 

XLI.     To  determine  a  mutual  inductance  in  terms  of 

a  capacity.    The  Carey  Foster  zero  method     178        259 

XLII.  To  compare  two  mutual  inductances.  Max- 
well's method 180  262 

XLIII.  Comparison  of  the  values  of  M  for  a  current 
inductor  as  determined  by  the  Carey  Foster 
zero  method,  and  by  direct  measurement  .  181  262 

* 

XLIV.  To  determine  a  mutual  inductance  with  the 
vibration  galvanometer  and  a  variable 
standard  of  self-inductance  .  .  .182  263 

XLV.  To  study  the  magnetic  leakage  about  a  mag- 
netic circuit 199  282 

XLVI.     To  find  the  inclination  with  the  dip  circle          .     202        287 
XLVII.     To  determine  Hby  the  magnetometer  method  .     204        294 

XLVIII.  To  determine  H  with  the  standard  tangent  gal- 
vanometer and  copper  voltameter  .  .  205  295 


xii  LIST  OF  LABORATORY  EXERCISES 

SEC.  PAGE 

XLIX.     To  compare  values  of  H  .         .         .        .         .         .206        297 

L.  To  determine  the  magnetic  quality  of  a  sample  of 
iron  by  the  ring-ballistic  method,  with  current 
reversals 223  326 

LI.     To  determine  the  B-H  curve  of  a  sample  of  iron 

with  the  Ewing  double  bar  and  yoke  method  .     227        335 

L1I.     To  determine  the  B-H  curve  of  a  sample  of  iron 

with  the  Fischer- Hinnen  traction  permeameter    230        340 

LIII.     To  determine  the  hysteresis  loss  in  a  sample  of 

iron.     Step-by-step  method       .         .        .         .233        345 

LIV.     To  determine  the  hysteresis  loss  in  a  sample  of  iron 

by  the  fixed  point  method          .        .         .         .235        360 

LV.     To  determine  the  flux  density  in  a  permanent  mag- 
net          236        353 

LVI.     To  determine  core-loss  with  the  Epstein  equipment    238        360 


ELECTRIC   AND   MAGNETIC 
MEASUREMENTS 


ELECTRIC    AND    MAGNETIC 
MEASUREMENTS 

INTRODUCTION 
PART  I.     LABORATORY  METHODS 

1.  Treatment  of  Errors  in  Laboratory  Work.  In  everyday 
life  it  is  not  uncommon  to  hear  positive  statements  as  to  the 
accuracy  of  frequently  measured  magnitudes,  such  as  mass  or 
length,  although  every  one  knows  that  instruments  of  increased 
precision  would  show  more  or  less  important  deviations  from 
the  stated  values.  In  accurate  scientific  work,  the  ideal  is  to 
make  measurements  which  shall  be  without  error.  This  ideal 
is  not  attainable,  however,  and  the  methods  chosen,  as  well 
as  the  values  observed,  must  be  studied  carefully  in  order  to 
determine  the  most  probable  value  of  the  quantity  which  ia 
being  measured.  In  every  measurement  attention  must  be 
paid  to  all  the  conditions  which  may  affect  the  correctness  of 
the  value  sought,  and  an  estimate  must  be  made  of  the  reliabil- 
ity, or  probable  precision,  or  trustworthiness,  of  the  final 
result. 

Let  us  suppose  that  a  galvanometer  deflection  of  8.55  cm. 
is  observed,  and  let  us  assume  that  a  careful  estimate  of  the 
probable  precision  of  the  measurements  leads  to  the  conclusion 
that  the  observer  can  be  sure  of  a  single  reading  to  within 
0.02  cm.  The  deflection  will  be  recorded  in  the  form 

d  =  8.55  ±  0.02  cm., 
B  1 


: . INTRODUCTION  [§  2 

which  means  that  the  recorded  result  will  not  be  in  error  by 
more  than  0.23  %. 

Errors  of  various  kinds  arise  in  making  measurements,  the 
most  important  of  which  are  the  following : 

Errors  of  observation,  due  to  lack  of  ability  on  the  part  of 
the  observer  to  make  accurate  readings. 

Simple  mistakes,  which  grow  less  troublesome  as  the  observer 
gains  in  training  and  experience. 

Instrumental  errors,  due  to  incorrect  calibration  of  scales  or 
to  faulty  adjustment  of  comparison  standards. 

Systematic  errors,  due  to  defects  in  the  method  used. 

It  is  readily  seen  that  the  effect  of  errors  of  observation  is 
reduced  by  taking  the  average  of  a  great  many  measurements, 
.in  which  the  chance  that  some  readings  are  too  large  is  offset 
by  the  chance  that  others  are  too  small.  Apparatus  of  a  higher 
grade,  more  carefully  made  and  calibrated,  is  important  for  the 
elimination  of  instrumental  errors,  while  repeating  the  experi- 
ment by  independent  methods  reduces  the  systematic  errors. 

2.  Laboratory  Methods.  In  order  to  accomplish  results  in 
the  laboratory  it  is  of  fundamental  importance  first  to  under- 
stand thoroughly  what  is  to  be  done,  and  then  to  proceed  in  a 
systematic  manner  to  do  it. 

In  general,  formulas  and  circuit  diagrams  should  not  be 
memorized,  but  instead  they  should  be  reproduced  as  the 
result  of  careful  study  and  logical  thought.  The  student 
should  be  able,  before  performing  the  experiment,  to  derive 
the  formulas  and  to  draw  the  circuits.  This  should  be  the 
outcome  of  a  thorough  understanding  of  the  argument  of  the 
problem,  however,  rather  than  definite  acts  of  memory. 

The  following  points  should  be  carefully  observed : 

(1)  Make  sure  that  you  know  fully  the  purpose  and  the 
theory  of  the  experiment,  including  the  meaning  of  all  the 
leading  terms. 


§  2]  LABORATORY  METHODS  3 

(2)  Be  sure  that  you  know  precisely  what  data  are  to  be 
secured.  v  . 

(3)  Prepare  in  advance  a  schedule  or  program  according  to 
which  the  observations  will  be  taken,  and  make  a  ruled  table 
with  headings  for  the  record.     This  makes  it  unlikely  that  any 
step  will  be  omitted,  and  furnishes  a  check  that  the  necessary 
data  are  in  hand. 

(4)  Secure  a  laboratory  notebook  or  observation  journal  of  a 
convenient  size,  preferably  permanently  bound,  and   keep   it 
exclusively  for  this  course,  dating  each  set  of  entries. 

(5)  Frequently  in  engineering  practice,  one  person  takes  the 
observations  and  another  person  reduces  and  discusses  them, 
perhaps  in  a  distant  city,  and  after  a  considerable  interval  of 
time.     This  renders  it  imperative  to  make  a  set  of  observa- 
tional data  complete  in  itself,  perfectly  clear  and  readable  and 
with  explanatory  notes  sufficiently  copious  so  that  the  influ- 
ence of  the  experimental  environment  may  .be  fully  grasped. 
The  student  should  keep  this  in  mind,  no  matter  how  simple 
may  be  the  problem  assigned  him. 

(6)  Scrutinize  the  conditions  under  which  the  observations 
are  taken  and  state  as  a  part  of  the  original  data  the  probable 
precision  of  the  single  readings  or  measurements. 

(7)  Proceed  systematically  in  making  the  required  adjust- 
ments.    The  student  will  be  greatly  aided   in   this  work  by 
arranging  the  circuits  in  a  neat  and  orderly  manner,  always 
avoiding  a  tangled  set  of  connecting  wires. 

Always  sketch  out  in  the  observation  journal  the  precise  cir- 
cuit as  it  was  used,  before  it  is  disconnected. 

(8)  The  report  should   contain  the  following  subdivisions : 

(a)  A  brief  statement  of  the  purpose  of  the  experiment. 

(&)  A  list  of  the  apparatus  used.  Each  piece  used  should  be  listed, 
together  with  the  maker's  name  and  the  number,  for  purposes  of  identi- 
fication, and  such  pieces  as  are  of  special  design,  or  of  exceptional  interest 
in  any  way,  should  be  described. 


4  INTRODUCTION  [§  2 

(c)  A  brief  analysis  of  the  entire  procedure,  which  will  include  a  con- 
cise review  of  the  whole  problem,  the  physical  principles  involved,  and 
the  fundamental  definitions. 

(d)  A  brief  statement  of  the  actual  steps  of  the  manipulation. 

(e)  A  table  of  data. 

(/)  Any  formulas  used,  together  with  the  meaning  of  each  symbol, 
and  a  sample  calculation.  The  derivation  of  the  formula  is  usually  neces- 
sary in  order  to  make  a  clear  statement  of  the  problem  as  required  under 
(c)  above. 

(0)  A  statement  of  the  degree  of  precision  attained  in  the  observations 
and  in  the  final  result. 

(A)  Do  not  omit  the  units  in  which  the  result,  and  any  other  impor- 
tant quantities,  are  expressed. 

(1)  Give  any  necessary  discussion  of  the  results  and  answer  any  ques- 
tions asked  in  the  text,  or  by  the  instructor. 

(9)  Precautions.  Bear  in  mind  that  the  apparatus  used  in 
the  electrical  laboratory  is  expensive  and  essentially  delicate. 
Handle  each  piece  with  the  utmost  care,  and  report  promptly 
any  accidents  which  may  occur.  In  using  electrical  measuring 
instruments  proceed  with  caution.  Use  the  lowest  shunt  first, 
or  the  highest  series  resistance,  or  the  scale  of  greatest  range, 
in  order  to  avoid  currents  too  large  for  the  apparatus.  In 
general  the  circuit  should  be  looked  over  and  approved  by  an 
instructor  before  final  connections  are  made.  The  connection 
to  the  source  of  power  should  always  be  the  last  one  made, 
and  this  should  be  done  cautiously.  The  importance  of  good 
contacts  must  not  be  overlooked.  The  ends  of  connecting 
wires  must  be  scraped  clean  and  joined  under  pressure  with 
double  connectors  or  in  binding  posts,  never  by  loosely  twist- 
ing them  together  with  the  fingers. 

Ordinarily  tap  keys  and  small  switches  are  only  intended 
for  use  with  feeble  currents  and  with  small  differences  of 
potential.  Sparking  and  consequent  oxidation  of  the  contact 
points  will  result  from  the  use  of  keys  or  switches  with  cur- 
rents which  are  beyond  their  intended  capacity. 


§  3]  DEFINITIONS  AND  UNITS  5 

PART  II.     FUNDAMENTAL  DEFINITIONS  AND  UNITS 

3.  Fundamental  Concepts  and  Definitions.  The  three 
quantities  upon  which  most  of  the  fundamental  relations  of 
electric  measurement  are  based  are  magnetic  pole  strength, 
magnetic  field  strength,  and  current  strength. 

The  unit  of  magnetic  pole  strength  is  that  strength  of  pole 
which  will  repel  a  similar  pole  of  equal  strength  at  a  distance 
of  one  centimeter,  in  air,  with  a  force  of  one  dyne. 

The  unit  of  magnetic  field  strength  is  that  field  in  which  the 
unit  pole  is  acted  on  by  a  force  of  one  dyne. 

A  magnetic  field  is  always  specified  in  terms  of  its  effect  on 
the  unit  pole.  The  direction  of  the  field  is  the  direction  in 
which  the  free  north-seeking  pole  will  move,  and  its  intensity 
is  measured  by  the  force  which  acts  upon  a  unit  pole  placed 
in  the  field. 

When  a  magnetic  pole  of  strength  ra  units  is  placed  in  a 
field  of  strength  H  units,  the  force  F,  in  dynes,  acting  on  the 
pole  is  given  by  the  formula 

F  =  mH  dynes. 

The  unit  of  current  strength,  in  the  electromagnetic  system, 
is  that  current  which,  flowing  through  a  conductor  placed 
normal  to  the  field,  will  experience  a  side  thrust  of  one  dyne 
for  each  centimeter  of  its  length. 

Direction  of  the  magnetic  field.  The  direction  of  the  mag- 
netic field  about  a  straight  wire  carrying  a  current  is  clockwise 
as  one  looks  along  the  wire  in  the  direction  of  the  current. 
In  a  circular  loop  or  solenoid  the  current  is  clockwise  to  an 
observer  looking  along  the  axis  in  the  direction  of  the  field. 

Reactions  between  magnetic  fields.  It  is  often  convenient 
to  regard  magnetic  fields  as  composed  of  discrete  lines  of  force. 
According  to  this  conception  the  following  rules  for  aiding 
the  memory  are  useful. 

Two  magnetic  fields   when   superposed    can   develop   force 


6  INTRODUCTION  [§  3 

actions  between  themselves  only  when  they  have  components 
which  are  parallel.  If  such  parallel  components  are  in  the 
same  direction,  the  fields  will  repel  one  another,  and  if  in 
opposite  directions,  they  will  attract  one  another.  In  repre- 
senting these  fields  011  paper,  the  plane  of  the  paper  must  con- 
tain these  parallel  components. 

Current  uniform  over  any  cross-section.  A  steady  current 
which  is  maintained  through  a  circuit  of  constant  resistance 
by  means  of  an  electromotive  force  has  the  same  value  through- 
out the  entire  circuit.  There  cannot  be  any  increase  or  de- 
crease of  current  strength  in  one  portion  of  the  circuit  as 
compared  with  any  other  portion. 

Ohm's  law  expresses  the  relation  which  exists  between  the 
three  fundamental  quantities,  current  strength  /,  electromotive 
force  or  potential  difference  E,  and  resistance  R.  It  may  be 
written  in  any  one  of  three  ways  : 


(2)  E  =  IR. 

(3)  /Z-|. 

Two  cases  of  the  application  of  this  law  will  be  considered  : 
first,  when  the  entire  circuit  is  taken  into  account,  and  second, 
when  only  a  portion  of  the  circuit  is  concerned. 

(1)  If  the  entire  circuit  is  considered,  the  value  of  E  in  the 
above  equations  must  be  the  maximum  potential  difference, 
or  the  E.  M.  F.  of  the  generator.  If  more  than  one  generator 
is  in  the  same  circuit,  E  will  be  the  algebraic  sum  of  all  the 
separate  electromotive  forces.  Similarly,  R  must  be  the  sum 
of  all  the  resistances  in  the  circuit,  including  the  internal  re- 
sistance of  the  generator,  or  of  all  the  generators,  if  there  is  more 
than  one.  For  this  case,  the  law  will  be  written  in  the  form 


§  4]  DEFINITIONS  AND  UNITS  7 

(2)  If  a  limited  portion  of  a  circuit  is  considered,  in  which 
there  is  no  generator,  the  value  of  E  in  the  above  equations  is 
the  potential  difference  impressed  across  _ 

the  terminals  of  the  constant  resistance  R    • 


F,G.  1.  * 
Any  variation  in  the  potential  difference 

is  accompanied  by  a  corresponding  variation  in  the  current 
strength,  and  the  ratio  E/I  is  constant,  and  always  equal  to  R. 

Potential  drop.  The  product  of  the  resistance  of  any  part 
of  a  circuit,  by  the  current  flowing  in  that  part,  equation  (2) 
above,  is  called  the  potential  drop,  the  fall  of  potential,  or  the 
IR  drop.  The  algebraic  sum  of  all  such  products  taken  around 
the  entire  circuit  is  then  equal  to  the  effective  or  resultant 
E.  M.  F.  in  the  circuit. 

Series  and  parallel  combinations  of  resistance.  When  sev- 
eral resistances  of  values  rlt  r2,  rs  —  are  connected  in  series, 
the  equivalent  resistance  of  the  group  is  given  by  the  formula 
R  =  n  +  r,  +  r3  .... 

When  several  resistances  of  values  rb  r2,  rz  •••  are  connected 
in  parallel,  or  in  multiple,  the  equivalent  resistance  of  the 
group  is  given  by  the  formula 

i    !+!+!.... 

R     r,     r2     r3 

4.  Establishment  of  Electric  Units.  For  measurements 
of  length,  mass,  and  time,  the  absolute  units  of  the  C.  G.  S. 
system  are  of  convenient  magnitudes.  However,  the  electro- 
magnetic units  derived  from  these  are  much  too  large  or  too 
small  for  practical  uses.  During  the  early  development  of 
electric  theory  the  units  were  in  a  confused  state,  frequently 
with  different  values  for  units  of  the  same  name. 

Several  international  conferences,  with  the  leading  physicists 
and  engineers  of  the  world  as  delegates,  were  called  from  time 
to  time.  At  the  International  Electrical  Congress,  held  in 


8  INTRODUCTION  [§  5 

Chicago  in  1893,  formal  definitions  for  the  principal  electric 
units1  were  adopted,  and  the  numerical  magnitudes  of  the 
ohm,  volt,  and  ampere  were  fixed.  All  of  these  units  were 
designated  as  the  International  Units.  These  units  were  made 
legal  in  the  United  States  in  January,  1894.  Since  that  time 
slight  changes  have  been  recommended  by  the  International 
Conference  at  London  in  1908,  and  by  its  authorized  commit- 
tee.2 The  units  as  adopted  by  this  Conference  are  defined  below. 

5.  Fundamental  Electric  Units.     The  ohm  is  the  unit  of 
resistance,  which  has  the  value  of  109  in  terms  of  the  centi- 
meter and  the  second. 

The  ampere  is  the  unit  of  current  strength,  which  has  the 
value  of  10"1  in  terms  of  the  centimeter,  gram,  and  second. 

The  volt  is  the  unit  of  electromotive  force,  which  has  the 
value  of  108  in  terms  of  the  centimeter,  gram,  and  second. 

The  watt  is  the  unit  of  power,  which  has  the  value  of  107  in 
terms  of  the  centimeter,  gram,  and  second. 

6.  The  International  Electric  Units.     In  order  to  repre- 
sent  these    fundamental    units    practically   for   purposes   of 
actual  measurement,  and  as  a  basis  for  legislation,  the  Confer- 
ence recommended  the  adoption  of  the  International  Units, 
defined  as  follows  : 3 

The  international  ohm  is  the  resistance  offered  to  an  un- 
varying electric  current  by  a  column  of  mercury 4  at  the  tem- 
perature of  melting  ice,  14.4521  grams  in  mass,  of  a  con- 
stant cross-sectional  area,  and  of  a  length  of  103.300  centi- 
meters. 

1  The  volt,  ampere,  ohm,  coulomb,  farad,  henry,  joule,  and  watt. 

2  For  details   of   this   Conference  see:    LONDON  ELECTRICIAN,  Vol.62, 
1908-09,    p.    104.     U.  S.  BUREAU  OF  STANDARDS:     (a)  Circular   No.    29, 
(b)  Miscellaneous  publications.     Report  of  the  International  Committee  on 
Electrical  Units  and  Standards. 

3  U.  S.  BUREAU  OF  STANDARDS,  Circular  No.  29. 

4  For  exact  specifications  see  Report  of  the  London  Conference,  LONDON 
ELECTRICIAN,  Vol.  62,  1908-09. 


§  6]  DEFINITIONS  AND  UNITS  9 

The  international  ampere  is  the  unvarying  electric  current 
which,  when  passed  through  a  solution  of  silver  nitrate  in 
water,1  deposits  silver  at  the  rate  of  0.00111800  gram  per 
second. 

The  international  volt  is  the  electric  pressure  which,  when 
steadily  applied  to  a  conductor  the  resistance  of  which  is  one 
international  ohm,  will  produce  a  current  of  one  international 
ampere. 

The  international  watt  is  the  energy  expended  per  second 
by  an  unvarying  electric  current  of  one  international  ampere, 
under  an  electric  pressure  of  one  international  volt. 

Based  upon  the  foregoing,  the  following  international  units 
are  readily  defined. 

The  joule,  equivalent  to  107  C.  G-.  S.  units,  is  the  work  done 
when  one  ampere  flows  for  one  second,  under  an  electric 
pressure  of  one  volt. 

The  coulomb,  equivalent  to  10"1  C.  G-.  S.  units,  is  the  quan- 
tity of  electricity  transferred  by  a  current  of  one  ampere  in 
one  second. 

The  farad,  equivalent  to  10~9  C.  G.  S.  units,  is  the  capacity 
of  a  condenser  which  is  charged  to  a  potential  of  one  volt  by 
one  coulomb. 

The  henry,  equivalent  to  109  C.  G.  S.  units,  is  the  inductance 
of  a  circuit  in  which  an  E.  M.  F.  of  one  volt  is  established  by 
a  current  varying  at  the  rate  of  one  ampere  per  second. 

In  addition  to  the  above  units,  there  were  adopted  by  the 
International  Convention  of  Electrical  Engineers  at  Paris  in 
1900  the  following : 

The  maxwell,  equivalent  to  one  C.  G.  S.  line  of  force,  is  the 
unit  of  magnetic  flux. 

The  gauss,  equivalent  to  one  maxwell  per  square  centi- 
meter, is  the  unit  of  magnetic  flux  density. 

1  For  exact  specifications  see  Report  of  the  London  Conference,  LONDON 
ELECTRICIAN,  Vol.  62,  1908-09. 


10 


INTRODUCTION 


:§7 


PART  III.    DIMENSIONS  AND  DIMENSIONAL  FORMULAS 

7.  Units  and  Dimensions.  In  order  to  define  any  physical 
quantity,  there  must  be  given  (a)  a  unit  and  (&)  a  numerical  coefficient 
which  shows  how  many  times  the  unit  is  repeated.  This  is  clearly  seen 
in  the  expression  of  such  quantities  as  10  pounds,  or  30  miles  per  hour. 

It  has  been  found  that  nearly  all  of  the  units  used  in  physics  can  be 
referred  back  to,  and  expressed  in  terms  of,  three  fundamental  units, 
length,  mass,  and  time.  The  powers  to  which  these  fundamental  units 
are  severally  raised  in  the  formula  for  any  derived  unit  are  called  the 
dimensions  of  that  unit.  For  example,  an  area  is  the  square  of  a  length, 
hence  the  dimension  of  area  is  2  as  regards  length.  Similarly,  a  volume 
has  the  dimension  3  in  length,  while  a  velocity  has  the  dimension  1  in 
length  and  —  1  in  time. 

In  writing  the  dimensions  of  units  attention  must  be  paid  to  the  de- 
fining equation  of  the  quantities  concerned.  In  the  table  below,  the 
dimensions  of  a  few  units  are  given,  together  with  the  defining  equations 
of  the  quantities. 


QUANTITY 

DEFINING  EQUATION 

DIMENSIONS 

Area        

a  =  l2 

H 

~~  t  ~  t2 

f=ma 
w=fl 

"7 

[XJ] 

[jfL2r-2] 

Volume   
Velocity  . 

Acceleration     

Force 

Work 

The  expressions  in  the  third  column  are  called  dimensional  formulas, 
and  it  is  customary  to  represent  the  three  fundamental  units  by  capital 
letters,  and  to  enclose  the  entire  group  in  square  brackets,  using  negative 
coefficients  where  units  enter  into  the  denominator  of  the  defining  equa- 
tions. These  formulas  indicate  how  the  units  of  the  various  quantities 
involve  the  three  fundamental  units  in  terms  of  which  they  are  defined, 
without  any  reference  whatever  to  the  numerical  coefficients  which  may 
be  associated  with  them.  These  numerical  coefficients  may  be  either  the 
numbers  which  represent  the  multiples  of  the  units  used,  or  they  may  be 


§  9]  DIMENSIONAL  FORMULAS  11 

quantities  like  TT,  or  the  trigonometric  functions.  In  any  case  such  numer- 
ical coefficients  have  no  dimensions,  and  do  not  enter  into  dimensional 
formulas. 

8.  Dimensional    Equations.      An    equation    signifies    that    the 
quantities  connected  by  the  sign  of  equality  must  be  alike  in  kind,  as 
well  as  identical  in  magnitude.     The  actual  numbers  substituted  in  for- 
mulas in   making   computations  are   the   numerical  coefficients,  but  all 
equations  must  be  carefully  examined  in  order  to  ascertain  whether  the 
dimensions  of  the  units  on  both  sides  of  the  equation  are  identical.     Such 
equations  with  their  various  qualities  expressed  in  terms  of  dimensions, 
as  shown  above,  are  called  dimensional  equations, 

9.  Uses  of  the  Theory  of  Dimensions.    The  study  of  the 

dimensions  of  physical  quantities  is  useful  in  three  ways  : 

(1)   To  check  formulas.     The  dimensional   formulas  may  be  used  to 

determine  whether  an  equation  is  homogeneous  with  respect  to  the  funda- 

mental units  involved  in  it. 

Consider  the  familiar  equation 


where  x  is  the  distance  passed  over  in  t  seconds,  by  a  particle  with  initial 
speed  s0,  and  a  constant  acceleration  a.  Written  in  dimensions  this 
becomes  - 


It  is  seen  that  x  is  the  sum  of  two  lengths.      Indeed,   quantities  of 
different  kinds  would  have  no  meaning  if  added  in  this  way. 
Again,  the  equation  for  the  period  of  a  simple  pendulum  is 


T=e< 

Written  with  regard  to  dimensions  only,  this  becomes 

In  these  examples  it  appears  that  every  term  in  any  one  of  the  equa- 
tions has  the  same  dimensions.  If  this  were  not  so,  it  would  be  proof 
that  there  was  some  error  in  the  formula. 

Dimensional  equations  do  not  check  the  correctness  of  the  numerical 
work  in  computing  ;  they  only  show  whether  the  reasoning  has  been 
correct  with  regard  to  the  fundamental  units  involved.  This  may  be 
further  illustrated  by  applying  dimensions  to  the  distinction  between 
mass  and  weight,  which  are  frequently  used  incorrectly  by  the  student. 


12  INTRODUCTION  [§  10 

Weight  is  a  force  and  hence  has  the  dimensions  [JtfZT-2],  while  mass 
is  itself  a  fundamental  unit  with  dimension  \_M~\. 

(2)  To  designate  unnamed  units.     It  is  frequently  convenient  to  use 
the  fundamental  units  as  they  appear  in  the  dimensional  formulas  when 
no  name  has  otherwise  been  given  to  a  unit.     Thus,  the  unit  of  speed  is 
given  as  foot/second  or  centimeter/second  ;   the  unit  of  acceleration  is 
foot/second2  or  centimeter/second2;   the  unit  of  moment  of  inertia  as 
gram-centimeter2  or  pound-foot2. 

(3)  To  find  the  new  value  of  the  numerical  coefficient  when  the  system 
of  units  is  changed.     If  n\  is  the  numerical  coefficient  of  a  given  quantity 
whose  unit  is  </i,  and  if  no  is  the  numerical  coefficient  of  the  same  quan- 
tity when  expressed  in  terms  of  another  unit  g2,  then 

mqi  =  W5#2 

or  <7i 

w2  =  ni  a*  • 

02 

For  example,  if  the  quantity  10  pounds  is  to  be  expressed  in  terms  of 
the  gram  as  a  unit,  then  712  equals  the  product  of  n\  by  the  ratio  of  the 
pound  to  the  gram,  and 

w2  =  10  x  453.6. 

As  another  example  of  the  usefulness  of  the  method,  we  may  find  what 
number  will  represent  33,000  foot-pounds  per  minute  when  C.  G.  S.  units 
are  used.  The  number  33,000  must  first  be  reduced  to  the  equivalent 
number  of  absolute  units  in  the  F.  P.  S.  system  or 

J-t^aa  x  32.1  =  550  x  32.1  foot-poundals  per  second. 
Then 


ni[  (gram)  (centimeter)2  (second  )~3]=  n2  [(pound)  (foot)2(second)~3], 

m  =  550  x  32.1  ["(PogndW       foot      Wgecond\-»-i 
L  \  gram  /  \centimeter  /  \second  /    J 

m  =  550  x  32.1[453.6  x  (30.5)2]  =  745+  x  107  ergs  per  second. 

10.  Dimensions  Of  Electric  Units.  There  are  two  systems 
of  electric  units,  the  electrostatic  and  the  electromagnetic,  based,  respec- 
tively, on  the  unit  charge  and  the  unit  magnetic  pole. 

Electric  charge.  The  fundamental  equation  in  the  electrostatic  system 
which  expresses  the  force  action  between  two  electric  charges  qi  and  #2, 
at  a  distance  apart  r,  is  given  by 

f-M*. 
kr* 


§  10]  DIMENSIONAL  FORMULAS  13 

In  this  equation  A:  is  a  constant  which  depends  upon  the  medium.  The 
dimensional  formula  for  charge  is  derived  in  the  following  way.  From 
the  above  equation 

qiqz  =  Fkr\ 


Magnetic  pole  strength.  The  fundamental  equation  in  the  electromag- 
netic system  which  expresses  the  force  action  between  two  magnetic 
poles,  mi  and  w2,  at  a  distance  apart  r,  is  given  by 


IT 

£    = 


In  this  equation  p  is  a  constant  which  depends  upon  the  medium.  The 
dimensional  formula  for  magnetic  pole  strength  is  derived  in  a  manner 
similar  to  that  for  charge  : 


In  the  above  equations  for  charge  and  magnetic  pole  strength  it  is 
difficult  to  understand  the  significance  of  the  fractional  exponents. 
Under  the  usual  assumption  that  k  and  fj.  are  both  unity  it  appears  that 
the  dimensions  of  m  and  q  are  the  same,  which  could  only  mean  that 
they  are  dimensionally  identical.  This  seems  peculiar,  and  the  expla- 
nation of  the  apparent  absurdity  lies  in  the  fact  that  both  k  and  /j.  may 
themselves  possess  dimensions.  But  since  we  do  not  know  the  precise 
mechanics  of  the  electrostatic  and  magnetic  phenomena  of  the  ether,  we 
cannot  write  the  dimensional  formulas  of  these  quantities.  If  we  could 
do  so,  the  fractional  exponents  might  be  rationalized.  Valuable  theo- 
retical results  follow  from  the  use  of  the  equations  containing  k  and  /*, 
but  the  usefulness  of  the  method,  so  far  as  the  immediate  applications  of 
dimensions  are  concerned,  is  not  impaired  by  omitting  them.  The  re- 
maining equations  will  be  derived  on  the  assumption  that  the  phenomena 
take  place  in  air,  in  which  case  k  and  ^  are  taken  as  equal  to  unity.  The 
dimensions  of  electric  quantities  based  on  the  electrostatic  system  will 
not  be  considered  further. 


14  INTRODUCTION  [§11 

11.  The  Electromagnetic  System.  In  deriving  the  dimensional 
formulas  based  on  the  electromagnetic  system  of  units,  the  starting  point 
is  always  the  force  action  associated  with  the  unit  magnetic  pole. 

Magnetic  field  strength.  When  a  magnetic  pole  of  strength  m  is  placed 
in  a  magnetic  field  of  strength  H,  the  force  action  is  given  by 

F=mH 
and 

*=*. 

m 

The  dimensional  formula  for  this  is 


H  =  =  [  jf 


Magnetic  flux.  Magnetic  flux  is  the  product  of  field  strength  by  the 
area  of  cross-section,  taken  at  right  angles  to  the  field,  or 

0  =  Ha. 
Hence  its  dimensions  are 

0  ^[jpzr^r-1!/2]  =  [jrWr-1]. 

Electric  current.  In  deriving  the  dimensional  formula  for  any  quantity, 
any  defining  equation  may  be  used,  so  long  as  the  dimensions  of  all  of  its 
involved  units  are  known,  except  the  one  to  be  derived.  In  the  case  of 
current,  we  may  start  with  the  force  on  a  conductor  of  length  Z,  placed  at 
right  angles  to  a  magnetic  field  of  strength  Jf,  and  carrying  a  current  of 
strength  f. 

F=iHl 
and 


The  dimensions  of  current  are  then 


Quantity  of  electricity.    The  quantity  of  electricity  which  passes  through 
a  circuit  with  a  constant  current  i,  in  time  <,  is  given  by 

Q  =  it. 
Its  dimensions  are 

Q  =  \_M^L^T-^T\  =  [JfW]. 

Potential  difference.    The  difference  of  potential  between  two  points  is 
measured  by  the  ratio  of  the  work  done  to  the  charge  carried,  or 

v=w. 

Q 

Its  dimensions  are 


§  11]  DIMENSIONAL  FORMULAS  15 

Resistance.  Resistance  may  be  defined  from  Ohm's  law,  in  terms  of 
current  and  potential  difference  : 

«=z. 

i 

Its  dimensions  are 

E  = 

Inductance.  Inductance  may  be  defined  in  terms  of  the  number  of 
linkings  per  unit  current,  where  N  is  given  by  the  product  of  the  magnetic 
flux  by  the  number  of  wire  turns  : 

j  _  N  _  n<f>  _  nHa 

~7~T~  ~T~ 

where  a  is  the  area  of  the  cross-section. 

Since  n  is  a  pure  number  it  need  not  be  further  regarded,  and  the 
dimensional  formula  of  inductance  is 


Capacity.  The  charge  in  a  condenser  is  given  by  the  product  of  the 
capacity  by  the  charging  potential  difference,  whence 

Q=CV 
and 

'-* 

Its  dimensions  are 

C  = 

The  student  is  not  advised  to  attempt  to  memorize  dimensional  formu- 
las. The  defining  equations  will  be  familiar,  however,  and  from  these  the 
dimensional  formulas  for  all  the  important  quantities  can  readily  be 
derived. 

EXERCISES 

1.  Find  the  number  of  joules  equivalent  to  100  foot-pounds. 

2.  A  mass  of  100  Ib.  moves  with  a  speed  of  500  ft.  per  second.     Find 
its  kinetic  energy  in  foot-pounds,  also  in  kilogram-meters. 

3.  Derive  the  dimensional  formula  for  inductance  from  the  intrinsic 
energy  equation. 

4.  Check  the  correctness  of  the   Helmholtz  equation  by  means  of 
dimensions. 

5.  Derive  the  dimensional  formula  for  resistance  from  Joule's  law. 

6.  Derive  the  dimensional  formula  for  potential  difference  from  the 
Faraday  equation. 


16  INTRODUCTION  [§  12 

PART  IV.     MISCELLANEOUS  INFORMATION 

12.  Prefixes.     Certain   prefixes   are   so  frequently  used  in 
scientific  work  that  they  should  be  quite  familiar.     The  more 
common  ones  with  their  meanings  are 

meg-  or  mega-  one  million 

kilo-  one  thousand 

hekto-  one  hundred 

deka-  ten 

deci-  one  tenth 

centi-  one  hundredth 

milli-  one  thousandth 

micro-  one  millionth 

13.  Keys  and   Switches.     It    is    frequently  desirable   in 
electric  work  to  open  or  close  a  circuit  quickly,  or  to  change 
the  connections  between  circuits  rapidly,  without  the  delay  or  in- 
convenience of  adjusting  wires  in  binding  posts.    Various  forms 

of  keys  and  switches  are  used  for  this 

e\     I        =  VT^     purpose.     The  simple  tap  key  is  shown 

_,  in  Fig.  2.     A  leaf  spring  is  held  rigidly 

at  e  on  a  base  6,  and  is  provided  with 

a  platinum  pin  at  d,  which  may  be  brought  into  contact  with 
a  platinum  surface  beneath  it  when  the  finger  is  pressed  on 
the  knob  c.  If  the  lower  contact  and  the  spring  at  e  are  con- 
nected to  a  pair  of  binding  posts  which  are  in  turn  connected 
to  any  circuit,  the  current  may  be  interrupted  or  established 
as  desired.  Platinum  contacts  are  necessary  in  order  to  pre- 
vent oxidation.  Such  keys  are  intended  for  use  with  feeble 
currents  and  low  voltage  only. 

A  switch  is  any  device  for  making,  breaking,  or  changing  the 
connections  in  an  electric  circuit.  A  single-pole,  single-throw 
switch  is  shown  in  Fig.  3,  and  in  any  given  case  one  will  be 


§13] 


MISCELLANEOUS  INFORMATION 


17 


a 


FIG.  3. 


chosen  of  sufficient  size  to  carry  safely  the  current  required. 

An  arrangement  of  two  such  switches  on  the  same  base,  with 

their  movable  blades  rigidly  connected 

by  an  insulating  bar,  is  called  a  double- 
pole,  single-throw  switch. 

If  the  blades  are  so  arranged  that 

they  may  be  thrown   clear  over,  and 

made  to  engage  jaws  on  the  opposite  side  of  the  rocking  point, 

it  is  called  a  double-pole,  double-throw  switch.  A  plan  view 
of  such  a  device  is  shown  in  Fig.  4.  By 
throwing  over  the  switch  blades  pivoted  at 
cd,  the  circuit  attached  to  the  terminals  cd 
may  be  connected  to  either  of  two  other 
circuits  connected  respectively  at  ab  or  ef. 

By  putting  cross  connections   between  the  terminal  posts  as 

shown  in  Fig.  5,  the  current  in  a  circuit 

connected  at  ef  may  be  reversed  through 

the    circuit    cd    by    throwing    the    switch 

blades  from  one  extreme  position  to  the 

other. 


CHAPTER   I 

GALVANOMETERS 

PART  I.     GENERAL  TYPES  AND  CHARACTERISTICS 

14.  Classification  of  Instruments.  Nearly  all  methods  of 
electric  or  magnetic  testing  require  the  measurement  of  one 
or  more  of  three  quantities,  current  strength,  potential  differ- 
ence, or  charge.  Frequently,  however,  the  problem  in  hand 
requires  only  that  the  presence  or  absence  of  current,  potential 
difference,  or  charge,  be  shown.  In  general,  essentially  the 
same  instrument,  with  certain  modifications,  may  be  used 
either  to  detect  the  existence  of  one  of  these  quantities  or  to 
measure  its  magnitude. 

An  instrument  for  simply  indicating  the  absence  of  a 
current  in  a  circuit  is  called  a  detector  or  an  indicating  gal- 
vanometer. 

When  used  to  measure  current  strength,  it  is  called  a 
current  galvanometer,  or  simply  a  galvanometer.  When  pro- 
vided with  a  scale  graduated  to  read  amperes,  it  becomes 
an  ammeter. 

When  arranged  to  measure  a  potential  difference,  it  is  called 
a  potential  galvanometer,  and  if  provided  with  a  scale  gradu- 
ated to  read  volts,  a  voltmeter. 

If  arranged  to  measure  quantity,  it  is  called  a  ballistic  galva- 
nometer. 

In  general,  the  term  galvanometer  is  reserved  for  a  class  of 
instruments  used  for  either  measuring  or  indicating  relatively 
feeble  currents  by  means  of  their  magnetic  effect. 

18 


I,  §  16]  TYPES  AND  CHARACTERISTICS  19 

15.  Galvanometer  Types.     Many  instruments  of  different 
kinds  have  been  designed   for  the   purpose   just  mentioned. 
Most  of  them  depend  upon  force  actions  between  magnetic 
fields.     Whatever  the  scale  of  the  instrument  reads,  or  what- 
ever the  quantity  to  be  measured,  the  observed  effect  is  really 
due  to  a  feeble  current  flowing  through  fixed  or  movable  coils, 
in  the  neighborhood   of   movable   or  fixed   magnets  or  other 
coils.     The  observed  motion  results  from  the   attraction  and 
repulsion  of  the  magnetic  fields.     One  member  of  the  system 
must  be   free  to  move   about  its  supports.     Two  types  are 
common,  the  suspended-needle    type  and   the   suspended-coil 
type. 

In  the  suspended-needle  type,  a  magnetic  needle  is  freely  sus- 
pended by  a  light  fiber  of  silk  or  quartz  at  the  center  of  a  coil 
of  wire  through  which  the  current  flows.  The  field  about  this 
coil  tends  to  set  the  needle  parallel  to  itself,  and  any  motion 
of  the  suspended  system  may  be  observed  directly,  or  more 
accurately  by  means  of  a  beam  of  light  reflected  from  a  small 
mirror  fastened  to  the  moving  part.  It  is  possible  to  so 
design  the  parts  that  the  deflections  are  very  nearly  propor- 
tional to  the  currents  passing  through  the  coils. 

In  the  suspended-coil  type,  a  permanent  magnet  is  fixed  in 
position,  and  between  its  poles  a  coil  of  wire  through  which 
the  current  is  to  pass  is  hung  by  a  thin  ribbon  of  phosphor 
bronze  or  steel.  A  helix  of  similar  material  attached  to  the 
bottom  of  the  coil  serves  as  a  flexible  connection  with  one  of 
the  coil  terminals,  the  other  being  attached  to  the  upper 
suspension  strip. 

16.  Torque.     Any  rotation  of  the  movable  parts  of  the  in- 
strument is  due  to  a  torque  or  turning  moment  applied  to  it. 
Figures  6  and  8,  pages  20  and  22,  show  respectively  how  the 
reaction  of  the  magnetic  fields  causes  the  torque  in  each  of  the 
two  types. 


20 


GALVANOMETERS 


(I,  §  16 


For  the  suspended-needle  type  (Fig.  6)  imagine  a  single 
circular  loop  of  wire,  whose  plane  lies  in  the  magnetic  merid- 
ian NS}  and  which  is  perpendicular  to  the  plane  of  the  paper 
intersecting  this  plane  in  A  and  B.  If  a  current  is  flowing 
clockwise  in  this  loop  as  seen  from  the  right-hand  side,  the 

N 
©A 


OB 

S 
FIG.  6. 

lines  of  force  will  be  shown  by  the  arrowheads  about  A  and 
B,  giving  a  resultant  field  FF  at  the  center  of  the  coil,  A 
magnetic  needle  originally  in  the  position  ns  will  be  deflected 
through  an  angle  6  to  n's',  at  which  position  the  moment  of 
the  deflecting  couple  is  just  compensated  by  the  moment  of 
the  restoring  couple.  Assume  the  strength  of  pole  of  the 
needle  to  be  m  units.  It  will  then  be  acted  upon  by  the  field 
F  with  a  force  of  Fm  dynes.  Similarly,  the  south  pole  is 
acted  upon  by  an  equal  and  opposite  force.  This  pair  of 
equal,  parallel,  and  oppositely  directed  forces,  not  in  the  same 
line,  constitutes  a  force  couple,  the  moment  of  which  is  given 
by  the  product  of  one  force  by  the  perpendicular  distance 


I,  §  16] 


TYPES  AND  CHARACTERISTICS 


21 


between  the  lines  of  action.  The  moment  of  this  deflecting 
couple  is 

(1)  L  =  Fml  cos  6. 

This  couple  will  deflect  the  needle  until  its  effect  is  overcome 
by  the  influence  of  the  magnetic  Held  (//)  in  which  it  swings, 
which  exerts  a  force  of  Hm  dynes  on  each  pole.  These  two 
restoring  forces  constitute  a  force  couple  of  which  the  mo- 
ment is 

(2)  L  =  Hml  sin  0. 

It  will  be  shown  in  Chapter  IV  that  the  magnetic  field  is 
directly  proportional  to  the  strength  of  current  in  the  coils.  J 
The  magnetic  field  //  is  the  resultant  field  due  to  the  horizon- 
tal component  of  the  earth's  field,  together  with  that  of  any 
control  magnet  which  may  be  in  use  with  the  galvanometer. 
Strictly,  the  torsion  of  the  suspension  fiber  is  also  acting 
against  the  deflecting  couple,  but  effort  is  made  to  reduce  this 


N 

£ 

£  L_ 

fs 

7> 

A  m  - 

1 

> 

t  ZJ 

FIG.  7. 

to  a  small  value  by  proper  selection  of  the  fiber.     It  is  con- 
sidered only  in  precise  work  of  a  certain  kind. 

For  the  suspended-coil  type  imagine  the  current  to  be  flowing 
down  in  the  right-hand  side  of  the  rectangular  coil  as  shown 
in  Fig.  7.  A  cross-section  of  Fig.  7  through  AB  is  shown  in 
Fig.  8,  page  22,  the  current  flowing  away  from  the  reader  at  a, 


22  GALVANOMETERS  [I,  §  16 

and  toward  the  reader  at  b.  The  arrows  show  the  direction  of 
the  magnetic  fields  about  each  coil  and  about  the  permanent 

magnet   NS.      Since 
magnetic  fields  in  the 
=5 —   same   direction  tend 
to  repel  one  another, 

while  those  in  oppo- 
FIG.  8. 

site   directions    tend 

to  attract,  the  resultant  torque,  due  to  the  reaction  of  the  two 
fields,  will  cause  the  coil  to  rotate  in  a  clockwise  direction. 
The  value  of  the  torque  is  found  as  follows.  Assume  the 
coil  to  be  I  cm.  long,  and  b  cm.  broad,  and  to  have  a  current 
of  /  C.  G.  S.  units  flowing  through  its  turns,  which  lie  in  a 
field  of  value  H  due  to  the  permanent  magnet.  The  side 
thrust  on  one  wire  in  the  field  H  is  given  by 

f=IHl  dynes 

and  on  one  side  of  the  coil  of  n  turns  it  is 
F  =  IHln  dynes. 

The  moment  of  this  force  is  (b/2)IHln.  Since  there  are  two 
sets  of  wires  corresponding  to  the  two  sides  of  the  coil,  the 
full  torque  L  is  given  by 

L  =  2-  IHln. 

2i 

The  product  bl  may  be  replaced  by  a,  the  area  of  the  coil, 
whence 

(3)  L  =  IHan. 

Equation  (3)  is  only  true  in  case  the  angle  0  is  very  small,  for 
as  rotation  takes  place  the  effective  lever  arm  is  shortened, 
and  the  value  of  the  torque  becomes 

(4)  L  =  IHan  cos  0. 


I,  §  18]  TYPES  AND  CHARACTERISTICS  23 

17.  Suspension  and   Control.     The  movable  part  of  the 
galvanometer  may  be  suspended  on  carefully  ground  pivots  in 
jewel  bearings,  or  in  the  sensitive  instruments,  by  means  of  a 
fiber  of  some  material  which  is  as  free  from  torsion  as  possi- 
ble.    Unspun  silk  is  frequently  used,  while  an  excellent  fiber 
can  be  made  by  drawing  into  a  fine  thread  a  bit  of  melted 
quartz.     These  quartz  fibers   can  be  drawn  exceedingly  fine, 
and  their  tensile  strength  is  relatively  high.     They  are  also 
free  from  troublesome  elastic  after  effects. 

In  the  case  of  the  moving  coil  instruments,  in  which  the 
suspension  must  carry  the  current  to  and  from  the  coil,  very 
fine  phosphor  bronze  or  steel  ribbon  is  used  above  and  a  spiral 
spring  of  similar  material  below.  These  ribbons  are  made  by 
rolling  a  wire  0.0015-0.0040  inch  in  diameter  into  a  flat 
ribbon. 

In  order  that  the  movable  system  may  return  to  its  position 
of  equilibrium  after  the  current  has  ceased  to  flow,  there 
must  be  a  controlling  torque. 

Referring  to  Fig.  6,  it  will  be  seen  that  the  control  is  here 
due  to  the  magnetic  field  in  which  the  needle  swings,  the  forces 
being  represented  by  the  vertical  arrows  at  the  poles  of  the 
needle.  In  the  moving-coil  instruments,  the  control  is  due 
entirely  to  the  torsion  in  the  suspending  metal  ribbon  and  the 
lower  spring. 

18.  Damping.     When  deflected  by  a  current,  the  suspended 
system,  controlled  as  it  is  by  a  restoring  torque  which  in- 
creases with  the  angle  of  deflection,  will  vibrate  with  progres- 
sively decreasing  amplitudes  about  its   final  position  before 
coming  to  rest.     Similarly,  when  the  current  ceases,  there  will 
be  an  oscillatory  motion  of  the  suspended  system  about  its  posi- 
tion of  equilibrium.     The  final  position  of  rest  is  only  reached 
after  all  the  energy  imparted  to  the  system  has  been  given  up. 
Since  it  is  tedious  to  wait  for  the  gradual  dying  away  of  this 
motion,  artificial  means  are  provided  for  absorbing  the  energy 


24 


GALVANOMETERS 


[I,  §  18 


in  order  to  bring  the  system  to  its  final  position  as  speedily 
as  possible. 

This  process  is  called  damping.  Damping  is  usually  accom- 
plished either  by  friction  of  the  air  against  a  light  vane 
attached  to  the  moving  parts,  or  by  the  generation  of  induced 


10  15  20  25 

Seconds 
FIG.  9. 

currents  due  to  the  relative  motion  of  conductors  and  mag- 
netic fields,  according  to  Lenz's  law. 

If  a  system  could  be  imagined  entirely  devoid  of  damping, 
it  would  continue  to  vibrate  indefinitely  after  a  displacement, 
as  shown  in  Fig.  9,  curve  1.  As  the  damping  increases,  the  pro- 
gressive decrease  in  amplitude  becomes  greater,  as  in  curve  2, 
until  finally  a  degree  of  damping  is  reached  for  which  the 


I,  §  19]  SPECIAL  TYPES  25 

motion  just  loses  its  periodic  character  and  becomes  aperiodic, 
as  shown  in  curve  3,  Fig.  9.  The  critical  value  of  the  damping 
for  which  this  transformation  takes  place  is  also  the  value  for 
which  the  return  of  the  system  to  its  zero  position  is  the 
quickest,  with  no  overswing  to  the  other  side.  As  the  damp- 
ing increases  beyond  this  critical  value,  the  return  to  zero 
becomes  slower,  and  this  condition  is  less  favorable  for  rapid 
work.  See  curve  4,  Fig.  9. 

19.  Mirror  and  Scale  Methods  of  Reading.  The  simple 
and  direct  method  of  attaching  a  light  pointer  to  the  moving 
part  of  the  galvanometer,  and  reading  the  deflection  as  this 
pointer  swings  over  a  graduated  circular  scale,  is  used  when 
great  accuracy  is  not  required,  and  when  the  moving  parts  are 
relatively  massive,  so  that  the  additional  mass  of  the  pointer 
is  negligible. 

For  sensitive  instruments  and  precise  work  a  small,  light 
mirror  is  fastened  to  the  moving  parts,  and  a  ray  of  light  is 
made  to  serve  as  a  pointer,  as  described  below. 

When  used  in  this  way,  the  instrument  is  called  a  reflecting 
galvanometer. 

The  lamp  and  scale.     In  Fig.  10,  SS'  represents  the  scaje 
at  a  distance  d  in  front  of  the  mirror,  which  is,  in  this  case, 
concave.     A  luminous   source,  such  as  the 
filament   of    an    incandescent    lamp,   or    a 
Nernst  lamp  glower,  is  mounted  at  L,  just 
above  or  below  the  scale.     The  parts  are 
adjusted  so  that  the  image  of  the  filament 
formed  by  the  mirror  shall  fall  at  the  zero 
of  the  scale,  and  if  the  scale  is  translucent, 
the  position  of  the  line  of  light  may  be  read  Flo 

from  the  side  opposite  the  mirror. 

The  telescope  and  scale.  In  Fig.  11,  SS'  represents  the 
scale,  T  the  telescope,  and  m  a  plane  mirror  fastened  to  the 


26 


GALVANOMETERS 


[I,  §  19 


moving  part  of  the  galvanometer.     The  portion  of  the  scale  at 
P  is  reflected  by  the  mirror  into  the  telescope,  the  objective 

lens  o  forming  a  real  image  of 
the  scale  at  x.  At  this  point 
are  placed  two  crossed  threads, 
usually  fibers  of  spider's  web, 
and  these  must  be  in  the  focal 
_jg  ie  plane  of  the  objective  o,  as  well 
as  of  the  eyepiece  e.  The  cross 
threads  will  then  be  clearly  seen 
superposed  on  the  image  of  the 
scale.  Any  motion  of  the  mirror 
will  then  cause  an  apparent 
displacement  of  the  scale  with  respect  to  the  cross  threads. 

If  a  ray  of  light  is  reflected  from  a  mirror,  making  an  angle 
0  with  a  normal  drawn  to  the  surface  of  the  mirror,  and  if  the 
mirror  is  rotated  through  some  angle  a,  it  can  be  shown  that 
any  change  in  «  produces  a  change  just  twice  as  great  in  0 
(Fig.  11).  It  follows  that  tan  2  a  =  tan  0  =  d/L,  where  L 
is  the  distance  from  mirror  to  scale.  Since  for  small  angles 
tan  2  a  may  be  put  equal  to  2  tan  «,  we  may  write 


FIG.  11. 


(5) 


tan  a 


d 
2L 


Hence  it  is  obvious  that  for  a  uniformly  graduated  straight 
scale,  the  scale  readings  are  not  proportional  to  the  angles  of 
deflection  of  the  mirror,  but  rather  to  tangents  of  twice  these 
angles.  For  very  small  deflections,  scale  readings  may  be 
assumed  proportional,  but  for  large  deflections,  greater  than 
about  five  degrees,  the  angles  should  be  computed. 

Some  galvanometers  have  curved  scales  of  constant  radius,  in 
which  case  scale  readings  are  proportional  to  angles  of  rotation. 

Frequently  the  galvanometer  is  used  in  zero  methods,  in  which 
case  a  short  straight  scale  with  arbitrary  graduations  is  sufficient. 


I,  §  21]  SPECIAL  TYPES  27 

PART  II.     DESCRIPTIONS  OF  SPECIAL  TYPES  OF 
GALVANOMETERS 

20.  Characteristics.     The  common  galvanometers  available 
for  use  in  the  laboratory  will  be  here  discussed  with  refer- 
ence to  the  five  points  mentioned  above :  (a)  how  the  torque 
is    set    up ;    (b)    suspension ;    (c)  control ;    (d)  damping ;    (e) 
method  of  reading. 

21.  Suspended  Needle  Galvanometers.    Two  examples  of 
this  type  will  be  considered. 

The  Thomson  or  Kelvin  galvanometer  is  named  from  Sir 
William  Thomson  (Lord  Kelvin),  who  perfected  it.  The 
torque  is  set  up  by  the  reaction  of  the  field  about  the  needle 
with  the  field  about  the  coils.  The  suspension  is  a  silk  or 
quartz  fiber.  The  control  is  chiefly  magnetic.  Damping  is 
effected  by  means  of  an  air  vane.  Readings  are  made  with 
the  lamp  and  scale,  or  with  the  telescope  and  scale. 

It  is  obvious  that  the  sensibility  increases  as  the  controlling 
force  is  weakened.  Hence,  to  reduce  the  control  as  much  as 
possible,  use  is  made  of  the  astatic  system.  This  consists  of 
two  similar  magnetic  needles  mounted  one  above  the  other  on 
the  same  support,  but  with  their  poles  reversed  in  direction. 
Surrounding  each  needle  is  a  coil  of  many  turns,  the  two  coils 
being  connected  in  series  and  with  the  direction  of  their  fields 
reversed.  With  this  arrangement  the  turning  effect  of  the 
feeble  current  is  doubled.  Were  the  two  needles  identical  as 
to  magnetic  moment,  the  effect  of  the  earth's  field  would  be 
exactly  neutralized,  and  the  system  would  stand  indifferently 
in  any  position.  If,  however,  there  is  a  slight  outstanding 
difference  in  the  magnetic  moments  of  the  two  needles,  which 
difference  can  be  made  as  small  as  desired,  the  controlling 
force  can  be  made  very  small. 

For  a  sensitive  galvanometer  of  this  type  it  is  necessary 
that  there  should  be  a  large  number  of  turns  of  wire  in  the 


28  GALVANOMETERS  [I,  §  22 

coils,  that  the  turns  should  be  close  to  the  needle,  that  the 
magnetic  poles  should  be  strong,  that  the  fiber  torsion  should 
be  small,  and  that  the  control  should  be  weak. 

The  tangent  galvanometer  takes  its  name  from  the  law  that 
the  current  strength  through  the  coils  is  proportional  to  the 
tangent  of  the  angle  of  deflection.  This  instrument  will  be 
described  further  and  its  theory  given  in  Chapter  IV.  The 
five  conditions  for  the  Kelvin  galvanometer  may  be  assumed 
to  hold  for  this  one  also,  except  that  the  control  is  chiefly  due 
to  the  earth's  field  alone. 

22.   Suspended-coil  Galvanometers.      D'Arsonval    Type. 

In  the  various  forms  of  instrument  of  this  type  the  torque  is 
due  to  the  reaction  of  the  field  about  the  movable  coil  with  the 
field  about  the  permanent  magnet/'  The  suspension  is  a  metal 
ribbon  above  and  a  coiled  spring  below,  both  serving  to  carry 
current.  Control  is  due  to  the  torsion  of  the  suspension. 

Damping  is  most  effectively  brought  about  by  induced 
currents.  These  may  arise  either  in  the  metal  frame  upon 
which  the  coil  is  wound  or  in  the  wire  turns  of  the  coil  itself. 
In  the  latter  case  a  tap  key  connected  across  the  terminals 
serves  to  short-circuit  the  coil  through  a  low  resistance.  If 
the  coil  is  formed  without  a  metal  frame,  a  closed  rectangle 
of  thick  copper  wire  may  be  attached  to  the  coil  by  spring 
clips.  As  the  coil  rotates  in  the  field,  current  is  induced  in 
these  moving  metal  parts,  and  the  associated  magnetic  field 
reacts  with  that  of  the  fixed  magnets.  The  forces  thus  called 
into  play  tend  to  oppose  the  motion  of  the  system.  The 
suspended -coil  galvanometer  is  practically  free  from  disturb- 
ances due  to  stray  fields  from  power  circuits  or  from  electric 
machinery. 

By  properly  shaping  the  pole  pieces  of  the  magnets,  and 
adding  a  cylindrical  soft-iron  core,  the  magnetic  field  in  the 
narrow  air  gap  in  which  the  coil  swings  may  be  made  uniform, 


I,  §  23]  SPECIAL  TYPES  29 

as  well  as  radial,  as  shown  in  Fig.  12.      In  this  case  the  de- 
flections of  the  coil  are  directly  proportional  to  the  current 
strengths  through  a  wide  range  of  angular  swing.     This  prin-x 
ciple  has  been  developed  and 
applied  in  the  direct-reading 
ammeter  and  voltmeter. 

Unless  extreme  care  is  taken 
during  the  process  of  maim- 
facture,  there  will  be  more  or 

less  magnetic  impurities  in  the  coil,  which  will  tend  to  give 
an  unsymmetric  deflection  on  the  two  sides  of  the  zero 
point.  In  general,  as  the  sensibility  of  the  galvanometer  is 
increased,  its  zero-keeping  quality  decreases.  Most  metal 
ribbon  suspensions  may  be  expected  to  show  a  slight  shift 
of  the  zero  after  a  full-scale  deflection.  Differences  in  deflec- 
tions on  the  two  sides  of  the  zero  are  due  chiefly  to  a  non- 
uniform  radial  field,  or  to  some  slight  displacement  of  the 
coil  from  its  symmetric  position  in  the  field. 

23.  Other  Types.  In  addition  to  the  galvanometers  de- 
scribed above,  there  are  many  others  of  special  design  for 
special  work.  We  shall  mention  only  four  of  these. 

Alternating-current  galvanometers  for  commercial  frequencies 
are  usually  of  the  electrodynamometer  type.  (See  Fig.  72.) 
The  action  depends  upon  the  reactions  of  magnetic  fields  in 
fixed  and  movable  coils. 

The  vibration  galvanometer  is  an  alternating-current  instru- 
ment which  depends  on  resonance.  It  will  be  described  in 
Chapter  IX,  in  connection  with  its  applications. 

High-frequency  galvanometers  are  used  for  measuring  alter- 
nating currents  where  the  frequency  may  vary  from  100,000 
to  2,000,000  cycles  per  second.  Such  instruments  are  in  con- 
stant use  in  wireless  telegraph  circuits,  and  they  must  be 
capable  of  measuring  a  wide  range  of  current  values.  These 


30  GALVANOMETERS  [I,  §  24 

values  may  be  as  great  as  200  or  300  amperes,  and  as  small  as 
a  few  milliamperes  or  even  a  few  microamperes. 

The  action  of  these  instruments  usually  depends  on  the  heat- 
ing effect  of  the  current,  which  may  be  measured  (a)  by 
expansion,  causing  a  stretch  or  sag  in  a  tense  wire ;  (b)  by 
thermoelectric  effect,  causing  a  deflection  on  a  sensitive 
potential  galvanometer ;  or  (c)  by  a  change  in  resistance, 
measured  by  means  of  a  Wheatstone  bridge. 

It  must  be  understood,  however,  that  measurements  of  the 
various  electric  quantities  of  these  high  frequencies  cannot 
be  carried  out  by  the  ordinary  methods  suitable  for  direct 
currents.  Satisfactory  calibration  of  such  instruments  is  only 
possible  in  specially  equipped  laboratories. 

The  Einthoven  string  galvanometer  consists  essentially  of  a 
fine  conducting  filament  carrying  current,  stretched  between 
the  poles  of  a  powerful  electromagnet.  When  current  passes 
through  the  filament,  it  is  displaced  transversely  by  the 
reaction  of  the  magnetic  fields,  and  this  displacement  is 
greatly  magnified  by  means  of  a  powerful  microscope.  This 
type  represents  the  highest  sensibility,  which  may  exceed 
that  of  a  good  d'Arsonval  by  3000  fold.  (See  §  29.)  A  cur- 
rent sensibility  of  10~13  amperes  per  scale  division  has  been 
obtained. 

24.  Galvanometer  Specifications.     The  complete  investi- 
gation of  any  galvanometer  involves  a  study  of  the  following 
factors :  (1)  principles  of  its  action,  (2)  diagrams  and  descrip- 
tions  of  its   parts,    (3)  resistance,    (4)  period,    (5)  damping, 
(6)  accuracy  of  its  return  to  zero  after  deflection,  (7)  sensi- 
tiveness, (8)  calibration  curves,  (a)  ballistic  constant  (Chap- 
ter VII). 

25.  Laboratory  Exercise  I.     A  study  of  galvanometer  types. 
Examine  carefully  such  galvanometers  as  may  be  available  in 


I,  §  25]  SPECIAL  TYPES  31 

the  laboratory,  especially  the  Kelvin  and  the  d'Arsonval  table 
and  wall  patterns. 

In  all  these  instruments  note  : 

(1)  General  type.  (5)  Method  of  suspension. 

(2)  Name  of  maker  and  serial     (6)  Method  of  control. 

number.  (7)  Method  of  damping. 

(3)  Exterior  appearance.  (8)  Method  of  reading,  plane 

(4)  Interior  appearance.  or  concave  mirror. 

In  the  report  discuss  the  various  instruments  examined  with  reference 
to  the  points  above  mentioned.  Make  sketches  showing  the  working 
features  of  the  type  forms. 

Using  the  hypothesis  of  lines  of  force,  make  drawings  to  show  how  the 
moving  system  is  turned.  State  whether  the  instruments  examined  are 
aperiodic,  and  discuss  the  conditions  which  affect  the  periodic  time  of  the 
suspended  system. 

State  the  conditions  which  govern  the  sensitiveness  of  the  different 
types. 


32  GALVANOMETERS  [I,  §  26 

PART  III.     SHUNT  CIRCUITS 

26.  The  Theory  of  Shunts.  It  frequently  happens  in 
electrical  testing  that  the  galvanometer  available  for  the  work 
in  hand  is  too  sensitive,  in  which  case  the  current  passing 
would  cause  a  deflection  beyond  the  range  of  the  scale,  and 
probably  damage  the  instrument.  A  convenient  and  much- 
used  method  for  safeguarding  the  galvanometer  is  that  of 
providing  a  shunt  or  by-pass  across  the 
terminals,  the  resistance  of  the  shunt  being 
so  adjusted  as  to  permit  more  or  less  of  the 
total  current  to  pass  through  it,  thus  leaving 
a  safe  and  measurable  fraction  of  the  total 
current  to  pass  through  the  galvanometer. 

Consider  a  galvanometer  of  resistance  g, 
provided    with     a    shunt    of    resistance    s 
(Fig.  13).     Let  /  be  the  total  current  flow- 
ing and  ig  and  is  the  currents  through  g  and  s,  respectively. 
Let  V  be  the  potential  difference  between  the  galvanometer 
terminals.     The  total  resistance  between  a  and  c  is  given  by 

(6)  *  =  ^-, 

g  +  * 

which  is  seen  to  be  a  fraction,  s/(g  +  s)  of  the  resistance  of  the 
galvanometer  alone. 

By  Ohm's  law  the  potential  drop  in  either  of  the  paths  is 

(7)  F=^  =  v, 
whence 


By  the  law  of  composition  in  proportion,  (8)  may  be  written 
(9)  _A_=^. 

*,  +  «.    g  +  s 

Remembering  that  ig  +  it  =  7,  (4)  becomes 


I,  §  26]  SHUNT  CIRCUITS  33 


(10) 
Then 


and 

(12)  1= 

S 

The  factor  (g  +  s)/s  is  called  the  multiplying  factor  or  multi- 
plying power  of  the  shunt,  because  it  is  the  quantity  by  which 
the  galvanometer  current  is  multiplied  in  order  to  give  the 
total  current  in  the  line.  B 

It  is  apparent  that  the  application  of  the  shunt  decreases 
the  resistance  between  a  and  6,  and  hence  the  total  current  /  is 
increased.  Hence,  the  total  current  is  not  a  constant  for  all 
values  of  s.  However,  the  relations  given  in  (7)  and  in  (12) 
are  perfectly  general  relations  between  the  galvanometer  cur- 
rent and  the  then  existing  total  current. 

In  order  to  study  the  effect  of  a  shunt  upon  the  total  current, 
it  is  important  to  examine  the  conditions  upon  which  the  value 
of  the  total  current  depends.  This  will  be  done  for  two  ex- 
treme cases :  I.  when  g  is  small  compared  to  the  total  circuit 
resistance ;  II.  when  g  is  large,  or  practically  the  only  resist- 
ance in  the  circuit. 

CASE  I.  When  g  is  small  compared  to  R.  Assume  a  circuit 
as  in  Fig.  13.  Let  E  represent  the  E.  M.  F.  of  the  battery,  b 
its  internal  resistance,  and  R  a  variable  control  resistance. 
Writing  Ohm's  law  for  the  entire  circuit  before  s  is  connected, 

(13)  I,  = 


Similarly,  after  s  is  connected 


34  GALVANOMETERS  [I,  §  26 

Dividing  (13)  by  (14),  we  have 

b  +  R  +  J!^ 

(15)  J*-  9  +  s 

I~     b  +  R  +  g 

The  condition  for  a  constant  total  current  is  that  the  ratio 
/i/Ja  shall  be  equal  to  1.  The  value  of  this  ratio  approaches 
unity  only  when  gs/(g  -f-  s)  approaches  g.  This  is  only  possible 
when  the  factor  s/(g  4-  s)  approaches  unity,  and  writing  this 
factor  in  the  form 


we  see  that  the  ratio  approaches  unity  as  g  decreases.  From 
equation  (15)  it  is  seen  that  as  g  approaches  zero,  the  ratio 
7i//2  approaches  unity,  and  it  is  then  apparent  that  when  g  be- 
comes negligibly  small  compared  to  the  rest  of  the  circuit  re- 
sistance, the  influence  of  the  applied  shunt  on  the  total  current 
is  also  negligible.  This  is  equivalent  to  saying  that  in  order 
to  keep  the  total  current  constant,  it  is  necessary  to  have  R 
very  large  as  compared  to  g.  This  case  arises  when  it  is 
desired  to  compare  values  of  the  currents  through  the  galva- 
nometer, with  and  without  the  shunt.  For  this  case  /  must 
not  be  appreciably  changed  by  the  application  of  the  shunt. 

From  equation  (12)  it  is  seen  that  the  ratio  of  total  current 
to  galvanometer  current  is  equal  to  the  multiplying  factor  of 
the  shunt.  If  the  desired  factor  is  n,  then 

Q  4-  s 
n  =  z-  -, 

s 
or 

ns  =  g  +  s, 
whence 

(16)  S  =  -L. 

n  —  1 

From  equation  (16)  it  is  seen  that  if  the  desired  multiplying 
factor  is  n,  the  necessary  shunt  value  is  determined  by  dividing 
the  galvanometer  resistance  by  n  —  1. 


I,  §  26]  SHUNT  CIRCUITS  35 

It  is  convenient  to  have  the  multiplying  factor  a  decimal 
multiple,  10,  100,  or  1000.  This  is  readily  attainable  if  s  be 
given  values  1/9,  1/99,  1/999,  respectively,  of  the  galvanome- 
ter resistance.  This  requires  a  shunt  box  or  set  of  shunt  coils 
especially  adjusted  for  the  galvanometer  with  which  it  is  to  be 
used.  In  the  Kelvin  type,  the  resistance  of  the  galvanometer 
coils  is  constant,  and  a  shunt  box  once  adjusted  is  thereafter 
trustworthy,  due  allowance  being  made  for  temperature 
changes  if  necessary. 

The  use  of  such  a  shunt  box  is  based  upon  the  assumption 
that  the  total  current  remains  constant.  This  assumption  may 
be  made  rigidly  true  if  series  resistances  are  introduced  into 
the  circuit,  of  such  values  as  to  compensate  for  the  decrease 
in  resistance  due  to  the  application  of  the  shunt.  Such  an 
arrangement  is  called  a  constant-current  shunt  box.  The  com- 
pensating-series  resistance  coils  are  automatically  added  to  the 
circuit  as  any  desired  shunt  is  applied  to  the  galvanometer. 

With  suspended-coil  galvanometers,  however,  the  resistance 
of  the  metal  suspension  is  considerable,  and,  inasmuch  as  it  is 
difficult  to  replace  a  suspension  without  altering  the  galva- 
nometer resistance,  a  satisfactory  adjustment  of  a  shunt  box 
is  impossible.  With  moving-coil  galvanometers,  modern  prac- 
tice makes  use  of  the  universal  shunt,  which  will  be  described 
in  §  27.  Its  use  will  be  illustrated  in  the  laboratory  exercise 
of  §  70. 

CASE  II.  When  g  is  large  compared  to  R.  In  this  case  we 
may  assume  that  R  -f-  b  is  negligible  as  compared  to  g  (Fig. 
13)  ;  that  is,  g  is  the  only  resistance  in  the  circuit.  Returning 
to  equation  (10)  of  Case  I,  we  may  neglect  J?  and  b  since  they 
are  very  small.  We  may  then  write 


L — ff  + * _   g 
^2~~    9    ~y  +  s 


36  GALVANOMETERS  [I,  §  26 

whence  we  have 

(17)  £  =  ^£±1. 

s 

From  (7),  however,  it  is  seen  that  the  total  current  is  the  gal- 
vanometer current  multiplied  by  (g  -f  s)/s,  whence 


(18) 


It  is  obvious  from  a  comparison  of  (17)  and  (18)  that  the  cur- 
rent ig  through  the  shunted  galvanometer  is  the  same  as  the 
original  current  /  through  the  galvanometer  before  the  shunt 
was  applied.  The  application  of  the  shunt  has  left  the  gal- 
vanometer current  unchanged.  This  means  that  in  this  case 
the  application  of  the  shunt  has  lowered  the  total  resistance 
of  the  circuit  and  thereby  increased  the  total  current  to  such 
a  degree  that  the  fraction  of  it  now  passing  through  the  gal- 
vanometer is  as  great  as  the  original  line  current.  In  this  case 
a  shunt  is  useless. 

To  illustrate  this  case  we  may  connect  a  high-resistance 
voltmeter  across  a  storage  battery.  A  shunt  of  any  value 
placed  across  the  terminals,  although  greatly  increasing  the 
current  drawn  from  the  battery,  has  very  little  effect  on  the 
current  through  the  voltmeter,  and  hence  its  reading  is  prac- 
tically unchanged. 

The  two  cases  considered  above  will  be  recognized  as  repre- 
senting extreme  conditions,  and  any  actual  case,  with  finite 
values  of  g,  s,  and  J?,  will  fall  somewhere  within  the  limits 
discussed.  The  actual  effect  on  the  total  current  for  any 
practical  case  can  be  ascertained  by  substituting  the  known 
values  in  equation  (15). 

EXERCISE 

Calculate  the  percentage  deviation  between  I\  and  1%  for  the  condi- 
tions &  =  1.0,  g  =  100,  s  =  10,  E  =  10  and  10,000. 


I,  §  27] 


SHUNT  CIRCUITS 


37 


27.  The  Ayrton  Universal  Shunt.  A  form  which  may  be 
used  with  any  galvanometer,  irrespective  of  its  resistance,  is 
the  so-called  universal  shunt  ^^ 

designed  by  Ayrton.  The 
complete  circuit  is  shown  in 
Fig.  15,  and  a  simplified  form 
for  explanation  is  shown  in 
Fig.  14.  In  Fig.  14,  the  gal- 
vanometer of  resistance  g  is 
permanently  shunted  by  a 
high  resistance  of  value  R. 


FIG.  14. 


The  battery  circuit  is  connected 
at  a  and  b,  b  being  a  mov- 
able contact  corresponding 
to  the  rotating  switch  arm 
in  Fig.  15.  The  resistance 
between  a  and  b  is  denoted 
by  r,  and  that  between  b 
and  c,  by  q.  Let  /  repre- 
sent the  total  current  flow- 
ing from  the  battery,  and 
assume  for  the  present 
that  it  is  constant  in  value, 
irrespective  of  variations 
in  r.  Let  iv  represent  the 
current  through  the  gal- 
vanometer when  b  is  at  c, 
and  i'2,  i'3,  etc.,  the  gal- 
vanometer currents  re- 
spectively for  different  positions  of  b.  From  equation  (11) 
we  have 


FIG.  15. 


and  this  may  be  adapted  to  the  circuit  of  Fig.  14  by  replacing 
s  by  r  and  g  by  (R  —  r  +  g).     Then  we  have 


38  GALVANOMETERS  [I,  §  27 


or 

(19)  <.= 

Assume  in  the  first  place  that  b  is  at  c,  then  (19)  becomes 

R 


(20)  t'i  =  / 


R  +  g 


Next,  assume  that   b   is  at  some  point   such  that  r  =  -fa  R. 
Then  (19)  becomes 

(2!)  *- 

Dividing  (21)  by  (20),  we  find 


whence 

(22)  h  =  Ah- 

Moreover,  since  galvanometer  deflections  are  proportional  to 
currents,  it  follows  that 

(23)  i2  =  4=J_. 
*!     di     10 

This  shows  that  the  ratio  of  iz  to  z'j,  or  that  of  the  currents 
through  the  galvanometer  for  the  two  cases  considered,  is  in- 
dependent of  the  resistance  of  the  galvanometer.  Hence  the 
same  ratio  holds  for  any  value  of  g.  It  is  necessary  here  to 
note  that  the  assumption  that  /  remains  constant  is  by  no 
means  always  true,  inasmuch  as  the  effective  resistance  in 
series  with  the  battery  is  varied  through  wide  limits  by 
changes  in  r.  However,  in  the  practical  applications  of  the 
shunt  box,  the  shunted  galvanometer  is  always  in  series  with 


I,  §  27] 


SHUNT  CIRCUITS 


39 


a  high  resistance,  of  the  order  of  T^  megohm  or  higher,  in 
which  case  any  change  in  r  would  have  very  little  effect  on  /, 
and  for  practical  purposes  .  ,  i 

1HH 


the  error  is  negligible. 

Applying  the  foregoing 
discussion  to  Fig.  15,  it  is 
seen  that  the  total  resist- 
ance of  4000  ohms  corre- 
sponds to  J?,  and  the  par- 
ticular fraction,  measured 
from  the  zero  end  to  the 
switch  contact,  corresponds 
to  r.  The  line  circuit  is 
connected  at  L  and  the 
galvanometer  at  G.  The 
five  sections  a,  b,  c,  d,  and 
e  have  a  total  resistance 
of  4000  ohms.  The  fol- 
lowing table  gives  the  ratios  for  the  various  positions. 


SECTION 

RESISTANCE 

SWITCH  POSITION 

RATIO  ill 

a 

0.4  ohm 

TtfW 

T^Tfatf 

b 

3.6  ohms 

TTjW 

TtfVff 

c 

36.0  ohms 

ifc 

Tfc 

d 

360.0  ohms 

TV 

A 

e 

3600.0  ohms 

1 

1 

Sometimes  it  will  be  convenient  to  use  the  box  for  obtain- 
ing a  fraction  of  any  available  voltage.  If  a  certain  voltage 
is  impressed  across  the  terminals  G,  there  will  be  available  at 
L  the  fractions  of  the  original  value  given  in  the  third  column 
of  the  table.  The  apparatus  may  then  be  used  as  a  volt  box. 
(See  §  99.) 


40 


GALVANOMETERS 


[I,  §  28 


28.  Laboratory  Exercise  II.  To  study  the  effect  of  shunts  on 
galvanometer  deflections. 

APPARATUS.  Battery  cell,  reversing  switch,  adjustable  re- 
sistance, resistance  box  with  low  values  for  s,  and  a  low- 
resistance  galvanometer. 

PROCEDURE.  (1)  Set  up  the  circuit  as  in  Fig.  16,  and  ad- 
just the  galvanometer  to  a  convenient  zero. 

(2)  With  K  open,  adjust  R  to  a  high  value,  making  the  de- 
flection ten  or  more  scale  divisions.  Read  deflections  on  both 
sides  of  the  zero  and  take  the  mean.  This  gives  a  mean  de- 
flection d,  which  is  proportional  to  the 
total  current  flowing. 

(3)  Keep  E  constant,  close  7T,  and 
adjust  s  until  the  observed  deflection 
d'   has    been   reduced   by  about  one 
tenth  of  that  first  read. 

(4)  Continue  reducing  s  by  similar 
steps,  about  ten  in  all,  until  s  becomes 
zero,  when  also  the  deflection  should 
be  zero.     The  connecting  wires  from  s 
to  the  galvanometer  should  be  as  short 

as   possible.      Reverse  each   time,   and    record    values   of  s, 
together  with  the  corresponding  mean  deflections. 

(5)  Tabulate   the   observed    data,   and  also    corresponding 
values  of  d'/d  and  s/(s-\-g).     State   in  the  report  what  the 
experiment  teaches. 

(6)  Plot  a  curve  with  values  of  s  as  abscissas,  and  deflec- 
tions as  ordinates,  and  state  what  inferences  may  be  drawn 
from  the   form  of  the   curve.     Draw  the  asymptote  to   the 
curve   corresponding   to   the  maximum  deflection.     Find  the 
value  of  the  shunt  such  that  any  desired  fraction  of  the  total 
current,  say  one  fifth,  flows  through  the  galvanometer. 


FIG.  16. 


I,  §  29]  CURRENT  SENSIBILITY  41 

PART  IV.     SENSIBILITY  OF  THE  CURRENT 
GALVANOMETER 

29.  Specification  of  Sensibility.  For  purposes  of  specifi- 
cation it  is  necessary  to  state  precisely  the  sensitiveness  of  a 
galvanometer.  Of  the  several  ways  in  which  this  may  be 
done,  the  following  are  the  most  useful : 

(1)  The   current  sensibility  is  the  current  strength  in  am- 
peres necessary  to  cause  a  deflection  of  one  millimeter  when 
the  scale  is  at  a  distance  of  one  meter  from  the  mirror.     This 
value  of  the  current  strength  is  called  the  figure  of  merit,  and 
galvanometers  ordinarily  in  use  have  values  ranging  from  10~4 
to  10~10.     Since  for  any  galvanometer  it  is  the  current  through 
its   coils,  together  with  the  associated  magnetic  field,  which 
causes  the  deflection,  the  figure  of  merit  is  the  fundamental 
definition  of  sensibility.     Knowing  the   applied  voltage   and 
the  circuit  resistances,  the  other  methods  of  specifying  sensi- 
bility may  be  derived  from  this. 

(2)  The   microampere   sensibility  is   the  number  of  milli- 
meters' deflection  caused  by  a  current  of  one  microampere 
when  the  scale  is  at  a  distance  of  one  meter  from  the  mirror. 

(3)  The   megohm   sensibility  is    the  number   of  megohms 
which  must  be  placed  in  series  with  the  galvanometer  in  order 
that  one  volt  shall  cause  a  deflection  of  one  millimeter  when 
the  scale  is  at  a  distance  of  one  meter  from  the  mirror.     If 
the   galvanometer   resistance  can   be   neglected  as   compared 
to  the  high-series  resistance,  which  is  usually  the  case,  it  will 
be   seen   that  the  megohm   sensibility  and   the   microampere 
sensibility  will  have  the  same  numerical  value. 

(4)  The  microvolt  sensibility  is  the  potential  difference  in 
microvolts  necessary  to  cause  a  deflection  of  one  scale  divi- 
sion.    This  is  chiefly  used  with  low-resistance  galvanometers 
and  those  of  a  portable  type  provided  with  pointers. 

(5)  The    voltage   sensibility  is   the   potential   difference   in 


42  GALVANOMETERS  [I,  §  30 

volts  necessary  to  cause  a  deflection  of  one  millimeter.  This 
value  will  always  be  the  product  of  the  resistance  of  the  gal- 
vanometer and  the  figure  of  merit. 

Since  sensibility  depends  on  the  number  of  wire  turns  in 
the  coil  or  coils,  which  in  turn  is  directly  proportional  to  the 
resistance,  it  follows  that  high-resistance  galvanometers  are 
usually  the  most  sensitive.  Portable  galvanometers  will 
usually  not  have  a  sensibility  greater  than  half  a  megohm. 
Wall  galvanometers  will  vary,  according  to  their  purpose, 
from  one  to  fifteen  hundred  megohms.  The  sensibility  will 
depend  upon  several  factors  such  as  position  and  strength  of 
control  magnet,  torsion  of  the  suspending  fiber,  number  of 
wire  turns  in  the  coils,  strength  of  the  magnetic  field,  and 
distance  from  the  mirror  to  the  scale.  Hence  a  value  of 
the  sensibility  of  any  instrument  must  be  accompanied  by 
a  precise  statement  of  the  conditions  under  which  it  is  de- 
termined. 

30.  Current  Sensibility  Formula.  The  formula  for  find- 
ing the  figure  of  merit,  or  the  current  sensibility  of  a  galva- 
nometer, is  derived  as  follows.  With  a  circuit  arranged  as  in 
Fig.  16,  K  being  open,  it  is  desired  to  find  what  current  will 
give  one  millimeter  of  deflection  for  some  stated  scale  dis- 
tance. Let  the  current  value  required  for  such  unit  deflection 
be  F  amperes,  let  E  denote  the  E.  M.  F.  of  the  battery,  and  let 
6  denote  the  internal  resistance  of  the  battery.  By  Ohm's 
law,  for  any  given  deflection  d,  we  may  write 

E 

whence 

(24)  F:  E 


In  general,  hoVever,  it  will  be  necessary  to  use  a  shunt  with 
a   sensitive  galvanometer,  which  may  be  done  by  closing  K 


I,  §  31]  CURRENT  SENSIBILITY  43 

and  adjusting  s  to  a  proper  value.  The  effective  resistance 
of  the  circuit,  including  shunt  and  galvanometer,  will  then  be 
sg/(s  -f-  g),  and  the  total  resistance  of  the  entire  circuit  will 
be  R  +  b  +  sg/(s  +  g).  Then,  if  the  E.  M.  F.  of  the  battery 
is  divided  by  this  total  resistance,  the  quotient  is  the  total 
current  in  amperes  flowing  through  the  circuit.  Since  the  cur- 
rent through  the  galvanometer  is  only  a  fraction  of  the  total, 
in  order  to  determine  the  current  effective  in  producing  the 
observed  deflection  d,  the  total  current  must  be  reduced  in 
the  ratio  s/(s+g),  as  in  equation  (11).  For  the  shunted  gal- 
vanometer equation  (24)  becomes 


(25) 


In  case  6  is  small,  as  it  usually  is,  it  may  be  neglected. 

31.  Laboratory  Exercise  III.  To  study  the  sensibility  of  a 
current  galvanometer. 

APPARATUS.  The  same  as  in  §  28.  Dry  cells  may  be  used 
if  a  sufficiently  high  resistance  is  put  in  series  with  them. 

PROCEDURE.  (1)  Connect  the  circuit  as  in  Fig.  16,  and 
adjust  the  galvanometer  to  zero  on  the  scale.  Note  carefully 
the  distance  of  the  mirror  from  the  scale,  as  well  as  any  other 
conditions  upon  which  the  figure  of  merit  may  depend. 

(2)  Keep  s  constant  and  adjust  It,  which  must  be  high,  so 
that  a  full  scale  deflection  is  read.     Increase  R  so  that  the 
deflection  is  reduced  by  about  one  fifth  of  the  first  value,  read 
both  right  and  left  deflections,  and  take  the  mean.     Continue 
increasing  R  for  five  or  six  values,  making  the  last  deflection 
about  two  centimeters. 

(3)  Calculate  the  values  of  F  for  each  mean  deflection,  and 
tabulate  values  of  E,  right  and  left  deflections,  mean  deflec- 
tions, and  F.     Record  also  the  values  of  6,  g,  and  E.     Reduce 


44  GALVANOMETERS  [I,  §  31 

the  deflections  to  millimeters.  If  the  scale  distance  is  not  one 
meter,  the  appropriate  correction  must  be  applied  to  the  values 
of  F.  In  case  F  is  found  to  vary  sensibly  from  a  constant, 
plot  values  of  current,  or  reciprocals  of  total  resistance  against 
deflections,  and  explain  the  significance  of  the  curves.  The 
galvanometer  resistance  g  may  be  found  directly  with  the 
Wheatstone  bridge,  or  by  the  half-deflection  method  of  §  55. 
Notice  that  different  procedure  is  necessary  according  to  the 
type  of  battery  used,  whether  non-polarizing  or  dry  cells.  The 
battery  resistance  may  be  found  by  the  half-deflection  method 
of  §  56.  The  E.  M.F.  may  be  read  directly  with  an  accurate 
voltmeter,  or  it  may  be  found  by  the  methods  described  in 
Chapter  III. 

(4)  From  the  mean  value  of  F  found,  calculate  the  sensi- 
bility of  the  galvanometer  according  to  the  definitions  (2),  (3), 
and  (5)  of  §  29.  Discuss  the  error  introduced  by  neglecting 
values  of  g  and  b. 

32.  Laboratory  Exercise  IV.  To  calibrate  a  galvanometer. 
In  many  experiments  in  which  galvanometers  are  used,  it  is 
necessary  to  know  whether  the  deflections  throughout  the 
entire  range  of  the  scale  are  strictly  proportional  to  the  cur- 
rents producing  them.  In  some  types,  currents  are  propor- 
tional to  sines  or  tangents  of  the  deflections,  while  in  most  of 
the  modern  instruments  the  effort  is  made  to  secure  strict 
proportionality  over  the  entire  scale. 

Arrange  a  circuit  as  in  Fig.  16,  and  observe  the  deflections 
for  ten  different  values  of  R. 

Plot  a  curve  between  values  of  the  deflections  and  recip- 
rocals of  total  circuit  resistance.  This  will  be  the  relative 
calibration  curve,  and  for  some  instruments  will  be  a  straight 
line.  If  the  actual  values  of  current  strength  are  plotted 
against  deflections,  the  absolute  calibration  curve  will  be 
traced. 


I,  §33]  POTENTIAL  GALVANOMETERS.  VOLTMETERS  45 

PART  V.     POTENTIAL  GALVANOMETERS.     VOLTMETERS 

33.  The  Potential  Galvanometer.  Deflections  of  a  gal- 
vanometer which  are  proportional  to  current  strengths  are 
also  proportional  to  the  potential  differences  impressed  at  the 
galvanometer  terminals,  since  the  resistance  of  the  instrument 
is  constant.  Such  galvanometers  may  then  be  used  to  meas- 
ure potential  differences,  and  they  are  called  potential  galva 
nometers,  or  voltmeters. 

The  potential  galvanometer,  or  voltmeter,  is  always  con- 
nected in  parallel  with  the  points  between  which  the  voltage 
or  potential  difference  is  to  be  measured.  It  is  readily  seen 
that  the  application  of  the  instrument  to  the  circuit  actually 
lowers  the  voltage  between  the  points  across  which  it  is  applied. 


FIG.  17. 

Assume  a  circuit  as  in  Fig.  17,  where  R  is  the  resistance  of  a 
portion  of  the  circuit  across  which  the  potential  galvanometer 
of  resistance  g  is  connected.  Let  Fi  be  the  potential  difference 
across  the  terminals  of  R,  before  the  galvanometer  is  applied. 
Let  V<2.  be  the  new  potential  difference  after  the  galvanometer  is 
connected.  Before  the  galvanometer  is  applied,  we  have 

(26)  F!  =  /!/?; 

and  after  the  galvanometer  is  applied,  we  have 

(27)  *-*A. 
Dividing  (26)  by  (27),  we  obtain  the  formula 


Rg  ' 


46  GALVANOMETERS  [I,  §  33 

or 
(28) 

In  order  to  insure  trustworthy  readings  of  the  potential 
galvanometer,  the  true  potential  difference  between  the  termi- 
nals of  R  must  not  be  appreciably  altered  by  the  application 
of  the  measuring  instrument,  and  hence  the  ratio  of  Fj  to  F2 
should  be  equal  to  unity.  In  any  actual  case,  however,  it  is 
readily  seen  that  the  potential  galvanometer  applied  as  a  shunt 


FIG.  18. 

across  the  terminals  of  R  actually  lowers  the  resistance  between 
these  points  and  correspondingly  increases  the  total  current 
flowing.  In  order  to  investigate  the  actual  change  in  the  value 
of  the  total  current  we  may  write  Ohm's  law  for  the  entire  cir- 
cuit, both  before  and  after  applying  the  potential  galvanometer. 
Referring  to  Fig.  18,  and  letting  E  represent  the  E.  M.  F.  of 
the  battery,  of  internal  resistance  6,  we  may  write 

(29)  /,  =  __!_, 
and,  similarly, 

(30)  /,  =  - 


E 


Dividing  (29)  by  (30),  we  have 


(31) 


/,_ 


R+g 


b  +  r 


I,  §34]  POTENTIAL  GALVANOMETERS.  VOLTMETERS  47 

Substituting  the  value  of  the  ratio  /i//2  as  given  by  equation 
(31)  in  equation  (28),  we  obtain  a  formula  for  the  ratio 
which  does  not  involve  /!  or  I2 : 


(32)  l 

F2 


=  bR  +  rR  +  bg  +  rg  +  Rg 
bg+rg  +  Eg 


bR  4-  Rr 

~r 


bg  +  rg  +  Rg 
or 

(33)  P  =  1  +  f 

"2  9       1      i 


From  (33)  it  is  seen  that  by  making  g  large  as  compared  to 
R,  F2  may  be  made  to  differ  from  V1  by  as  small  an  amount 
as  may  be  desired.  A  necessary  condition  for  a  potential  gal- 
vanometer is  that  its  own  resistance  shall  be  high.  The  actual 
value  which  g  must  have  in  order  to  secure  a  required  mini- 
mum variation  in  F2  as  compared  to  V\  is  readily  computed 
from  equation  (33),  if  the  other  circuit  resistances  are 
known. 

In  practical  voltmeters  the  value  of  the  internal  resistance 
is  usually  made  not  less  than  100  ohms  for  each  volt  of  scale 
range. 

34.  The  Voltmeter  Multiplier.  It  is  often  desired  from 
the  standpoint  of  economy  or  convenience  to  make  a  potential 
galvanometer  or  voltmeter  available  over  a  range  greater 
than  that  of  its  own  scale. 


GALVANOMETERS 


[I,  §  34 


a 

t 

d       j 

! 

*i 

/^N                   M 

'  (Vm\  k/^/VV^ 

FIG.  19. 


Assume  that  the  range  of  the  voltmeter  Vm  (Fig.  19)  is  3 
volts  for  a  full  scale  deflection,  and  that  it  is  desired  to  measure 
with  it  a  value  of  30  volts.  Since  the  maximum  deflection  is 
caused  by  a  potential  difference  of  3  volts  between  its  terminals 

6c,  the  voltmeter  obviously 
cannot  be  connected  di- 
rectly across  ad,  for  the 
potential  difference  be- 
tween these  points  is  30 
volts.  However,  if  a  re- 
sistance of  value  M  is 
placed  in  series  with  the 
voltmeter  where  M  is  so 
chosen  that  the  potential  drop  through  it  is  27  volts,  then 
the  full  scale  (or  3-volt  point)  may  be  marked  30,  and  a  ten- 
fold increase  in  scale  range  will  always  be  indicated  so  long 
as  this  particular  resistance  is  used  in  series  with  the  instru- 
ment. 

Such  a  resistance  is  called  a  multiplier.  It  may  be  separately 
mounted  in  a  suitable  case,  or  it  may  be  inclosed  within  the 
case  of  the  voltmeter,  its  terminals  being  so  arranged  that  it 
can  be  put  in  series  with  the  voltmeter  when  the  larger  range 
is  desired. 

The  value  of  M  is  readily  derived  as  follows.  When  the 
voltmeter  of  resistance  g  is  connected  as  shown  in  Fig.  19,  a 
feeble  current  i  flows  through  it.  Then  the  potential  drop  in 
it  alone  is  given  by 


(34) 


Vl  =  ig  =  3  volts. 


The  potential  drop  through  the  voltmeter  and  the  multiplier 
together  is  given  by  the  equation 


(35) 


F2  =  i 


=  30  volts. 


I,  §  35]  POTENTIAL  GALVANOMETERS.  VOLTMETERS  49 

If  the  value  of  the  current  is  the  same  in  both  cases,  so  that 
the  same  deflection  of  the  pointer  is  caused,  we  may  divide 
(34)  by  (35)  and  cancel  t,  whence 

(36) 

This  reduces  to 


Thus,  if  the  resistance  of  the  voltmeter  is  300  ohms,  M  = 
2700  ohms  for  a  tenfold  increase  in  scale  range. 

In  general,  if  it  is  desired  to  increase  the  scale  range  by 
some  factor  n,  the  resistance  of  the  multiplier  will  always  be 
the  voltmeter  resistance  multiplied  by  (n  —  1).  The  student 
should  compare  this  case  with  that  of  the  galvanometer  or 
ammeter  shunt,  in  which  case  the  galvanometer  resistance  is 
divided  by  (n  —  1). 

A  desirable  uniformity  follows  from  the  practice  of  making 
the  voltmeter  resistance  100  ohms  for  each  volt  of  scale  range, 
in  which  case  multipliers  are  interchangeable. 

35.  Voltmeters  Used  to  Measure  Current.  It  has  been 
pointed  out  that  a  necessary  condition  for  sensibility  in  galva- 
nometers is  that  the  movable  system  must  be  light.  In  the 
suspended-coil  type  of  instrument  this  means  that  only  very 
fine  wire  may  be  used,  and  the  current-carrying  capacity  is 
therefore  very  small. 

On  this  account  an  ammeter  seldom  carries  through  its 
movable  coil  the  total  current  to  be  measured.  However,  it  is 
really  designed  as  a  potential  galvanometer,  to  show  the  fall  of 
potential  across  an  accurately  known  low  resistance  or  shunt, 
through  which  most  of  the  current  passes. 

Since  for  a  constant  resistance  the  potential  differences  are 
directly  proportional  to  the  current  strengths,  it  follows  that 


50  GALVANOMETERS  [I,  §  35 

the  scale  of  the  instrument  may  be  calibrated  to  read  directly 
in  amperes. 

The  usual  arrangement  is  shown  in  Fig.  20,  where  A  is  the 
sensitive  potential  galvanometer.  The  deflections  are  propor- 
tional to  the  potential  differences  between 
the  ends  of  S,  and  the  scale  is  graduated 
to  read  appropriate  units  of  current 
strength  in  the  line. 

A  potential  galvanometer  may  be  pro- 
vided with  several  low-resistance  shunts  of  different  values. 
It  is  then  available  for  reading  current  strength  over  any  de- 
sired range.  These  shunts  may  be  contained  within  the  case 
of  the  instrument,  or  they  may  be  separately  mounted  outside 
the  case,  and  provided  with  current  and  potential  terminals. 

It  is  not  practicable  to  use  interchangeable  shunts  with  low- 
resistance  galvanometers.  Consider  an  ammeter  of  0.01  ohm 
resistance  shunted  with  a  resistance  of  equal  amount.  The 
current  through  the  ammeter  is  half  the  total  current,  hence 
the  scale  range  is  doubled.  However,  if  the  shunt  is  remov- 
able or  interchangeable,  the  contact  resistances  are  likely  to  be 
too  large  to  be  neglected.  The  contact  resistance  of  a  number 
16  wire  clamped  under  a  binding  post  with  moderate  pressure 
will  not  be  less  than  0.0001  ohm,  and  may  be  as  great  as 
0.001  ohm.  In  this  case  the  contact  resistance  is  at  least  one 
per  cent,  and  may  be  ten  per  cent,  of  the  value  of  the  shunt, 
and  the  effective  value  of  the  shunt  is  too  great  by  this 
amount.  Hence  it  is  necessary  to  make  the  resistance  of  the 
galvanometer  high,  and  use  it  as  a  potential  galvanometer.  In 
this  case  the  small  contact  resistances  are  negligible. 

36.  Laboratory  Exercise  V.    Some  experiments  on  Ohm's  law. 
APPARATUS.     Voltmeter,  ammeter,  resistance  box,  dry  cells, 
single  resistance,  and  tap  key. 

PROCEDURE.     (1)  Connect  the  voltmeter  across  the  battery 


I,  §36]  POTENTIAL  GALVANOMETERS.  VOLTMETERS  51 

terminals,  and  record  the  reading.  Insert  in  series  with  the 
voltmeter  and  battery  a  resistance  box,  and  adjust  this  until 
the  reading  is  reduced  to  exactly  one  half  its  former  value. 
Write  Ohm's  law  for  each  case,  and  solve  for  the  voltmeter 
resistance. 

(2)  Calculate  the  value  of  the  multiplier   necessary  to   in- 
crease the  scale  range  of  the  voltmeter  tenfold.     Check  this 
value  by  actual  trial  with  the  resistance  box. 

(3)  Connect  the   battery   in   series  with  the   ammeter,  the 
single  resistance,  and  the  tap  key.     Connect  the  voltmeter  in 
parallel  with  the  resistance.     Read  and  record  several  simul- 
taneous values  of  current  and  voltage  and  compute  the  value 
of  the  resistance. 

(4)  Connect  the  ammeter  and  voltmeter  in  series  with  the 
battery,  and  record  the  reading  of  each.     Explain  the  result. 

Remember  that  the  ammeter  is  a  low-resistance,  direct- 
reading  current  galvanometer.  The  voltmeter  is  a  high- 
resistance,  direct-reading  potential  galvanometer. 

QUESTIONS.  Two  ammeters  are  connected  in  series  with  a 
battery.  Will  the  readings  be  alike  or  different?  Two  volt- 
meters are  connected  in  series  with  a  battery.  Will  they 
read  differently  or  the  same :  (a)  if  they  have  the  same  re- 
sistance? (6)  if  they  have  different  resistances?  What  is 
the  effect  of  too  great  a  resistance  in  an  ammeter  ?  What  is 
the  effect  of  too  small  a  resistance  in  a  voltmeter  ? 


EXERCISES 

1.  Assume  a  circuit  like  that  in  Fig.  18,  and  let  E  =  10,  b  =  2,  and 
r  —  10.     It  is  desired  that  the  voltmeter  readings  shall  not  be  in  error 
by  more  than  one  half  of  one  per  cent.    Find  the  value  of  the  voltmeter 
resistance. 

2.  How  can  a  line  voltage  of  90  volts  be  measured  with  two  voltmeters, 
each  of  which  has  a  range  of  50  volts  ?    The  resistances  of  the  voltmeters 
are  4000  ohms  and  6000  ohms  respectively. 


52  GALVANOMETERS  [I,  §  36 

3.  Two  voltmeters  of  150  volts  range  and  15,000  ohms  resistance  are 
placed  in  series  across  a  110-volt  circuit.     What  will  be  the  readings  on 
each  of  the  two  instruments  ?    Suppose  that  the  resistances  of  the  volt- 
meters are  11,500  ohms  and    16,000  ohms   respectively.     What  current 
flows  through  each  instrument,  and  what  is  the  voltage  reading  on  each  ? 

4.  How  can  a  current  strength  of  90  amperes  be  measured  with  two 
ammeters,  each  of  which  has  a  range  of  0-50  amperes  ?     Find  the  current 
through  each,  and  the  potential  difference  at  the  terminals  of  each,  (a)  if 
the  resistances  are  each  equal  to  0.05  ohm  ;  (6)  if  the  resistances  are 
0.045  and  0.055  respectively. 

5.  An  ammeter  and  a  voltmeter  are  to  be  used  for  measuring  a  re- 
sistance by  Ohm's  law.     They  may  be  connected  as  shown  in  either  (A) 


-(^m) 'WW^ 


FIG.  B. 

or  (.B).    Discuss  the  relative  merits  of  these  two  arrangements,  especially 
when  E  has  (a)  a  high  value  ;  (&)  a  low  value. 

6.  A  4-volt  battery  has  an  internal  resistance  of  1  ohm.     It  is  put  in 
series  with  a  resistance  R  of  3  ohms,  across  which  is  connected  a  volt- 
meter of  3000  ohms  resistance.     Find  (a)  the  total  current ;  (6)  and  (c) 
the  potential  difference  across  R  before  and  after  the  voltmeter  is  con- 
nected ;  (d)  the  percentage  error  in  the  voltmeter  reading. 

7.  In  order  to  secure  a  multiplying  power  of  ten,  a  galvanometer  of 
90  ohms  resistance  is  shunted  with  10  ohms.     Assume  an  error  of  ±  £  % 
in  the  shunt.    What  is  the  effect  on  the  value  of  the  multiplying  power  ? 

8.  A  voltmeter  of  3000  ohms  resistance  and  an  ammeter  of  0.2  ohm 
resistance  are  connected  in  series  with  a  battery  of  3  volts.     Neglecting 
the  internal  resistance  of  the  battery,  find  (a)  the  current  which  flows 
in  each  instrument;    (6)   the   potential  drop  in  the  voltmeter;  (c)  the 
potential  drop  in  the  ammeter. 


CHAPTER   II 

RESISTANCE  AND   ITS   MEASUREMENT 
PART  I.     GENERAL  DEFINITIONS 

37.  Resistance  and  Ohm's  Law.  The  relations  existing 
in  an  electric  circuit  in  which  a  steady  current  is  flowing  be- 
tween electromotive  force,  current,  and  resistance  were  first 
accurately  studied  by  G.  S.  Ohm.  He  stated  these  relations  in 
a  work  which  he  published  in  1827.1 

The  law  of  the  electric  circuit  which  expresses  these  rela- 
tions is  called  Ohm's  law  (§  3).  This  law  states  the  experi- 
mental fact  that  the  strength  of  current  is  always  directly 
proportional  to  the  electromotive  force,  and  inversely  propor- 
tional to  the  resistance.  It  may  be  written  in  the  form 


where  /  is  the  strength  of  current,  E  is  the  electromotive  force, 
and  R  is  a  quantity  called  the  resistance,  which  is  character- 
istic both  of  the  material  of  which  the  circuit  is  made  and  of 
the  form  (length  and  cross-section)  in  which  this  material  is 
disposed.  It  is  found  that  the  resistance  is  not  dependent 
upon  the  strength  of  current  flowing  nor  upon  the  electro- 
motive force.  It  depends  only  upon  (a)  the  nature  of  the 
material,  (6)  the  form,  size,  and  physical  state  of  the  material, 
(c)  the  temperature.  If  the  electromotive  force  is  doubled, 
the  character  of  the  circuit  remaining  otherwise  unchanged, 
the  current  is  found  to  be  doubled  ;  in  other  words,  the  ratio 

1  Die  Galvanische  Kette,  Mathemattsch 
53 


54        RESISTANCE  AND  ITS  MEASUREMENT     [II,  §  37 

E/I  is  constant  and  equal  to  R.  The  fact  that  this  ratio 
remains  constant  shows  that  R  corresponds  to  a  definite  phys- 
ical property  of  the  material.  The  resistance  may  then  be 
defined  as  that  characteristic  constant  of  the  circuit  which  is 
the  constant  ratio  of  E  to  /. 

The  basic  idea  underlying  the  use  of  resistance  in  electric 
circuits  is  that  of  opposition  to  the  flow  of  current.  If  the 
resistance  is  altered,  the  current  strength  is  altered  in  an  in- 
verse ratio.  This  opposition  to  the  flow  of  current  is  analo- 
gous to  friction  in  mechanics,  and  it  is  always  accompanied 
by  the  production  of  heat.  If  an  unvarying  current  is  main- 
tained through  a  metallic  conductor  by  a  constant  potential 
difference,  the  electrical  energy  is  entirely  transformed  into 
heat,  the  amount  of  which  is  proportional  to  the  resistance. 
The  rate  of  transformation  of  energy  is  given  by  Joule's  law 
(Chapter  IV,  §  111)  : 


where  i  is  the  current  strength,  R  is  the  resistance,  H  is  the 
heat  produced,  and  t  is  the  time. 

It  follows  that  resistance  may  be  measured  by  means  of  the 
heat  generated,  as  well  as  by  the  ratio  of  simultaneous  values 
of  E  and  /.  This  ratio,  and  hence  the  value  of  the  resistance, 
remains  constant  only  so  long  as  the  rate  of  dissipation  of  the 
heat  produced  is  equal  to  the  supply  rate  of  the  electrical 
energy,  so  that  the  temperature  of  the  conductor  does  not 
change. 

In  applying  Ohm's  law  to  any  portion  of  a  circuit,  which 
does  not  contain  an  electromotive  force,  the  current  is  given 
by  the  ratio  of  the  potential  difference  impressed  on  the  ter- 
minals of  the  resistance  to  the  resistance  itself.  If  the  law  is 
applied  to  the  entire  circuit,  the  total  resistance  of  the  circuit 
must  be  used,  including  the  internal  resistance  of  the  generator, 
or  of  all  the  generators  if  there  are  several.  Likewise,  the 


II,  §  38]  GENERAL  DEFINITIONS  55 

resultant  or  net  value  of  all  the  electromotive  forces  must  be 
used.  If  the  generators  are  in  series,  this  resultant  value  is 
found  by  taking  the  algebraic  sum  of  all  the  electromotive 
forces  present  in  the  entire  circuit. 

In  the  case  of  some  substances,  such  as  electrolytes,  in 
which  polarization  phenomena  appear,  there  are  in  general,  in 
addition  to  the  potential  difference  impressed  at  the  elec- 
trodes, internal  potential  differences  which  must  be  taken  into 
account  before  applying  Ohm's  law.  These  internal  potential 
differences,  which  are  opposed  to  the  impressed  potential  dif- 
ference, will  virtually  have  the  effect  of  a  resistance,  and  care 
must  be  taken  that  the  actual  resistance  found  has  been  freed 
from  these  effects. 

It  has  been  found  that  the  ratio  of  potential  difference  to 
current  is  not  exactly  constant  for  many  non-metallic  sub- 
stances. This  is  true  for  gases,  and  also  for  such  substances 
as  rubber,  paraffin,  and  shellac.  The  same  departure  from  a 
constant  ratio,  especially  for  high  voltages,  is  found  for  many 
high-resistance  liquids,  such  as  certain  oils,  benzine,  and  ether. 
In  these  instances  it  is  probable  that  polarization  phenomena 
are  present,  as  in  electrolytes.  It  is  also  probable  that  actual 
changes  in  the  value  of  the  resistance  are  caused  by  the  appli- 
cation of  the  electromotive  force. 

With  the  corrections  indicated  above,  Ohm's  law  has  been 
verified  after  most  careful  investigation  for  currents  passing 
through  metals  and  electrolytes,  and  with  a  precision  of  one 
part  in  one  hundred  thousand.  Hence  this  law  is  a  general 
principle  of  fundamental  importance  for  electrical  measure- 
ments. 

38.  Units  of  Resistance.  The  absolute  unit  of  resistance 
may  be  derived  from  Joule's  law.  It  is  that  resistance  in 
which  the  C.  G.  S.  unit  of  current  strength  will  dissipate  one 
erg  of  heat  in  one  second.  This  is,  however,  not  easily  real- 


56        RESISTANCE  AND  ITS  MEASUREMENT     [II,  §  38 

ized  experimentally,  nor  is  it  of  convenient  size.  A  multiple 
of  109  is  chosen,  which  is  practically  represented  by  the  resist- 
ance of  a  mercury  column  at  0°  C.,  106.300  cm.  long,  of  mass 
14.4521  grams,  and  of  constant  cross-section.  This  practical 
unit  is  the  international  ohm  (§  5). 

For  purposes  of  convenience,  resistances  are  grouped  in 
three  ranges,  according  as  their  values  are  medium,  high,  or 
low.  Medium  resistances  are  ordinarily  measured  in  ohms, 
and  include  values  from  about  1  ohm  to  100,000  ohms. 
Values  beyond  this  range  are  called  high;  they  are  most  con- 
veniently expressed  in  megohms  (§  12).  Fractions  of  an  ohm, 
especially  small  fractions,  are  called  low,  and  are  most  con- 
veniently expressed  in  microhms  (§  12). 

39.  Resistivity,  or  Specific  Resistance.  Experiment  has 
shown  that  the  resistance  of  a  conductor  whose  cross-section 
is  uniform,  and  whose  substance  is  homogeneous,  is  directly 
proportional  to  its  length,  inversely  proportional  to  the  area 
of  its  cross-section,  and  is  also  proportional  directly  to  some 
constant  which  has  different  values  for  different  materials. 
If  R  is  the  resistance  of  a  piece  of  wire  of  cross-section  a  and 
length  /,  then  we  may  write 

(2)  R  =  k1-, 

where  k  is  a  constant  depending  on  the  material.  Solving 
this  equation  for  Tc,  we  find 

(3)  *=:£?. 

It  follows  that  for  a  conductor  of  unit  length  and  unit  cross- 
section,  7c  is  equal  to  R ;  hence  Jc  may  be  defined  as  the  resist- 
ance between  opposite  faces  of  a  unit  cube  of  the  substance. 
This  value  k  is  called  the  resistivity  or  the  specific  resistance 
of  the  material.  Since  the  resistance  of  a  sample  unit  cube 


II,  §  39]  GENERAL  DEFINITIONS  57 

is  always  small,  it  is  usually  expressed  in  terms  of  microhms 
per  unit  cube.  Instead  of  giving  resistivity  in  terms  of  re- 
sistance, unit  length,  and  unit  cross-section,  it  may  be  ex- 
pressed also  in  terms  of  the  resistance  in  ohms  per  foot  of 
wire  one  mil  (0.001  inch)  in  diameter,  or  the  resistance  in 
ohms  per  meter  of  wire  one  millimeter  in  diameter. 

Another  group  of  units  of  resistivity  is  based  upon  the  mass 
of  the  sample  instead  of  the  volume ;  for  example,  the  resist- 
ance of  a  piece  of  uniform  wire  which  has  a  length  of  one 
meter  and  a  mass  of  one  gram,  or  the  resistance  of  a  piece  of 
wire  which  has  a  length  of  one  mile  and  a  mass  of  one 
pound.  The  units  of  resistivity  may  be  summarized  as 
follows : 

microhm  per  centimeter3, 


Volume  units 


I  ohm  per  meter-gram, 
Mass  units 


ohm  per  mil-foot, 
I  ohm  per  millimeter-meter. 

j  ohm  per  meter-gram, 
(  ohm  per  mile-pound. 


In  practice,  the  mass  units  of  resistivity  are  preferable  to  the 
volume  units  for  the  following  reasons :  (a)  the  measurement 
of  the  cross-section  is  frequently  difficult  or  inaccurate,  (&)  for 
many  shapes  this  measurement  is  impossible,  (c)  copper  is 
ordinarily  sold  by  mass  rather  than  by  volume  and  hence  the 
data  of  greatest  value  are  given  more  directly. 

The  mass  units  of  resistivity  may  be  readily  converted  into 
volume  units  if  the  density  of  the  sample  is  known.  For 
standard  annealed  copper  at  20°  C.,  the  density  is  8.89.  A 
wire  of  this  material  one  meter  long  and  one  square  millimeter 
in  cross-section  has  a  resistance  of  0.1724  ohm.  If  the  wire 
is  one  meter  long  and  one  gram  in  mass,  its  resistance  is  0.15328 
ohm.1  The  figures  will  vary  slightly  for  different  samples.  A 

1  Much  data  of  value  will  be  found  in  Copper  Wire  Tables,  U.  S.  BUREAU 
OF  STANDARDS,  Circular  No.  31. 


58        RESISTANCE  AND  ITS  MEASUREMENT     [II,  §  39 

certain  sample  of  copper  has  the  following  values  for  its  re- 
sistivity when  expressed  in  the  various  units  : 

900.77      pounds  per  mile-ohm, 
0.1577  ohm  per  meter-gram, 
1.7726  microhms  per  centimeter3, 
10.663    ohms  per  mil-foot. 

40.  Conductivity.     The    reciprocal    of    the    resistivity    is 
called  conductivity  and  its  unit  is  the  mho,  or  reciprocal  ohm, 
per  centimeter  cube.     For  many  reasons,  especially  in  engineer- 
ing practice,  conductivity  is  specified   instead  of   resistivity. 
Apparatus  of  the  bridge  type  has  been  arranged  for  measur- 
ing directly  in  terms  of  arbitrary  standards  the  conductivity 
of   such  materials  as  copper  rods  and  wires,  and  steel  rails. 
The  samples  used  must  be  short  for  reasons  of  economy,  and 
the  methods  of  measurement  follow  closely  those  given  below 
for  low  resistances. 

41.  Temperature  Coefficient  of  Resistance.     The  electrical 
resistance  of  all  substances  is  found  to  change  more  or  less 
with  changes  in  the  temperature.     All  pure  metals  and  most 
alloys  have  their  resistance  increased  with  rising  temperature, 
while  carbon  and  many  electrolytes  show  a  decrease  in  resist- 
ance with  increasing  temperature. 

Experiment  has  shown  that  the  resistance  of  any  conductor 
at  a  temperature  t°  C.  is  given  by  the  formula 

(4)  Rt  =  R,  +  R,at  +  R£V+~., 

where  R0  is  the  resistance  of  the  sample  at  zero,  a.  is  the 
change  in  1  ohm  when  the  temperature  changes  frtom  0°  to 
1°  C.,  and  /?  is  a  measure  of  the  variation  per  degree  in  a. 
The  factor  a  is  called  the  temperature  coefficient.  The  value 
of  ft  is  very  small ;  for  copper  it  is  0.0000012 ;  hence  it 


II,  §  42]  GENERAL  DEFINITIONS  59 

may   be   neglected   in  all   but  the   most  precise   work.     The 
simplified  formula  for  metallic  conductors  will  then  read 


(5)  Rt 

The  temperature  coefficient  is  a  function  of  many  factors, 
such  as  the  process  of  purifying  the  metal,  the  degree  of  its 
purity,  and  the  methods  of  drawing  and  annealing  during 
manufacture.  All  pure  metals  have  nearly  the  same  value  of 
the  temperature  coefficient,  approximately  0.4  of  one  per  cent 
per  degree  C.  If  values  of  resistance  are  plotted  against 
corresponding  temperatures  on  the  absolute  scale,  the  curves 
approach  straight  lines,  with  a  general  trend  toward  the  origin, 
indicating  that  at  the  absolute  zero  the  conducting  metals  lose 
their  resistance.  It  has  recently  been  shown  by  Onnes,  in 
experiments  with  conductors  surrounded  by  helium  boiling 
under  diminished  pressure,  that  at  a  temperature  of  three  or 
four  degrees  above  the  absolute  zero  the  resistance  does  prac- 
tically disappear.  In  this  state  Ohm's  law  no  longer  holds, 
and  when  current  flows  through  the  conductor  at  this  temper- 
ature, there  is  neither  a  generation  of  heat  nor  a  fall  of 
potential. 

42.  The  Formula  for  Temperature   Coefficient.    If   the 

values  of  resistances  of  a  conductor  at  two  temperatures  other 
than  zero  are  known,  the  value  of  a  may  be  derived  in  the 
following  way.  Let  t  and  t'  be  two  temperatures  for  which 
the  corresponding  values  of  resistances  are  Rt  and  Rt>.  Then, 
from  equation  (5),  we  may  write 

(6)  Rt  =  R0(l  +  «*), 

(7)  R,  = 
Dividing  (6)  by  (7),  we  find, 

(8)  A  = 


60        RESISTANCE  AND  ITS  MEASUREMENT     [II,  §  42 
Clearing  of  fractions  and  solving  for  a,  we  have 


43.  Current  Control  by  Means  of  Resistance.    One  of  the 

most  frequent  needs  in  the  electrical  laboratory  is  the  control 
of  current  strength,  and  this  is  readily  accomplished  by  means 
of  resistances,  either  fixed  or  variable  in  value,  included  in  the 
circuit.  The  variable  resistances  are  grouped  under  two 
heads,  resistance  boxes  and  rheostats.  A  third  group  includes 
standard  coils  of  fixed  values.  The  common  resistance  ma- 
terials and  the  various  controlling  devices  above  mentioned 
will  be  taken  up  in  order. 

44.  Resistance  Materials.     For  standard  coils  and  resist- 
ance boxes  it  is  important  that  the  material  used  should  have 
the  following  qualities  : 

(a)  Permanence,  so  that  a  coil  once  adjusted  to  a  given  value 
may  be  relied  on  for  a  long  period  of  time  ; 

(b)  Small  temperature  coefficient,  so  that  changes  in  tempera- 
ture may  affect  the  resistance  in  a  small  degree  ; 

(c)  Large  resistivity,  so  that  a  high  resistance  may  be  assem- 
bled without  too  great  bulk  ; 

(d)  Small  thermoelectric  effect  against   copper  or  brass,  so 
that  variations  in  temperature  between  different  parts  of  the 
circuit  may  not  cause  troublesome  thermal  currents. 

Laboratory  standards  of  precision  are  sometimes  made  of 
mercury  in  glass,  but  these  are  not  readily  portable,  and 
they  are  exceedingly  fragile,  requiring  the  highest  technical 
skill  in  their  assembly  and  use.1 

Early  materials  for  resistance  coils  were  german-silver 
and  platinum-silver  alloys,  but  these  had  a  high-temperature 
coefficient.  Hence,  other  alloys  were  sought  which  would 

1  See  U.  S.  BUREAU  OF  STANDARDS,  Bulletin,  vol.  12,  p.  375. 


II,  §  44] 


GENERAL  DEFINITIONS 


61 


more  nearly  meet  all  the  conditions  mentioned  above.  A  large 
number  of  copper-nickel  alloys  have  been  produced  which  are 
useful  because  they  have  high  resistivity,  and  in  general,  low- 
temperature  coefficients ;  but  they  are  not  available  for  pre- 
cision work  because  of  their  high  thermoelectric  effects.  In 
the  alloy  called  manganin,  however,  this  objection  is  almost 
entirely  overcome,  as  the  thermoelectromotive  force  between 
manganin  and  copper  does  not  exceed  1.5  microvolts  per 
degree  C.  The  table  herewith  given  shows  the  composition  of 
some  of  the  most-used  resistance  materials,  together  with  the 
resistivity  and  temperature  coefficient. 


PROPERTIES  OF  RESISTANCE  MATERIALS  l 


Substance 

Composition 

Resistivity 
microhms—  cm*. 

Temperature 
Coefficient 

Aluminum     

3.2 

0.0039 

Copper  (annealed)  .     .     . 

1.7 

0.0042 

Iron  (pure)     

9.96 

0.0062 

wire 

10.0-15.0 

/    20°  C.  0.0052 

\  800°  C.  0.015 

Steel  soft-hard    .... 

15.9-46.0 

0.0042-0.0016 

transformer  plate      . 

11.1 

silicon-steel     .     .     . 

61.0 

0.004-0.006 

Mercury 

95.8 

0.00088 

{Cu  1 

la  la     

60.0 

0.00001 

Ni   -     j 

f  Cu  84%  ] 

Manganin 

|  Ni    12%  V 

42-47 

12°  +  0.0001 

1  Mn    4%  j 

30°  _  0.00002 

f  Cu  50%  "I 

German  silver     .... 

|  Zn  30%  i 

30.0 

0.0033 

[  Ni   20%  j 

Arc  lamp  carbon     .     .     . 

4000-6000 

-  0.0003 

1  Complete  tables  of  data  on  these  properties  will  be  found  in  the  various 
electrical  handbooks ;  also  in  LANDOLT-BORNSTEIN,  Physikalisch-Chemische 
Tabellen. 


62        RESISTANCE  AND  ITS  MEASUREMENT     [II,  §  45 

45.  Resistance  Boxes.  Resistance  boxes  are  groups  of 
coils  of  wire  arranged  compactly,  so  that  single  coils,  or  any 
series  combination  of  them,  may  be  introduced  into  the  circuit 
by  manipulation  of  switches  or  plugs.  These  coils  may  range 
in  value  from  a  few  tenths  or  hundredths  of  an  ohm  up  to  a 
tenth  of  a  megohm.  They  are  usually  non-inductively  wound 
with  fine  wire  of  small  current-carrying  capacity.  They  are 
used  only  with  feeble  currents,  usually  small  fractions  of  one 
ampere. 

What  constitutes  a  safe  current  for  any  particular  box  can 
only  be  determined  by  some  knowledge  of  the  way  the  box  is 
made,  and  the  size  of  the  wire  used  in  its  coils.  Usually  the 
maker  selects  the  wire  so  that  the  rate  of  transformation  of 
electric  energy  into  heat  is  practically  constant  for  all  the 
coils,  and  the  carrying  capacity  is  usually  rated  in  terms  of 
the  watts  dissipated,  without  overheating. 

The  watt  load  on  any  coil  is  readily  found  from  the  formula 
EZ/R,  where  E  is  the  voltage  across  the  coil  and  R  is  its  re- 
sistance. The  current,  impressed  voltage,  and  total  circuit  re- 
sistance must  be  so  adjusted  that  the  safe  minimum  for  the 
box  in  hand  shall  not  be  exceeded.  In  general,  this  will  be 
about  one  watt,  although  some  resistance  boxes  on  the  market 
will  dissipate  four  watts  without  overheating.  The  common 
type  of  wood  spool  boxes  would  have  the  paraffin  coating  soft- 
ened at  two  watts ;  complete  safety  requires  that  the  tempera- 
ture shall  not  rise  more  than  a  few  degrees  above  that  of  the 
room. 

In  any  event,  especially  if  a  storage  battery  or  dynamo  is 
the  source  of  power,  some  estimate  should  be  made  of  the 
electromotive  force  across  the  coils,  and  of  the  probable  resist- 
ances in  circuit,  and  the  current  should  be  so  controlled  that 
the  power  supply  does  not  exceed  one  watt,  unless  the  reasons 
for  allowing  larger  values  are  approved  by  those  in  charge  of 
the  laboratory. 


II,  §  45] 


GENERAL  DEFINITIONS 


63 


The  accuracy  of  adjustment  of  the  coils  in  resistance  boxes 
varies  greatly  with  the  kind  of  box  selected.  For  most  pur- 
poses one  fifth  to  one  tenth  of  one  per  cent  is  sufficient.  If 
greater  precision  is  desired,  the  coils  are  wound  on  metal 
spools  and  the  whole  inclosed  in  a  metal  case  with  perforated 
sides,  to  permit  the  use  of  an  oil  bath  whose  temperature  may 
be  controlled  accurately. 

In  order  to  vary  the  number  of  coils  which  are  at  any  time 
included  in  a  circuit,  some  form  of  rotary  dial  switch,  or  sim- 
ple plug  switch  will  be  used.  The  latter 
is  illustrated  in  Fig.  21.  xWhen  the  plug 
P  is  withdrawn,  current  can  pass  through 
the  coil  r,  which  is  connected  to  the  brass 
blocks  a  and  b.  With  the  plug  inserted 
the  coil  is  short  circuited  by  a  path  of 
practically  zero  resistance. 

The  use  of  plugs  is  legitimate  only  when 
they  fit  accurately  into  their  sockets,  and 
when  the  contact  surfaces  are  clean.  The  contact  surface  of 
the  plugs  should  not  be  touched  with  the  hand,  nor  should  the 
plugs  be  laid  carelessly  on  the  work  table.  Idle  sockets  are 
sometimes  provided  to  receive  the  plugs  when  not  in  use; 
otherwise  they  should  be  laid  on  a  sheet  of  clean  paper,  or 
placed  in  a  clean  box  kept  for  that  purpose.  A  thin  film 
due  to  oxidation  tends  to  prevent  perfect  metallic  contact, 
and  a  slight  twisting  motion  is  desirable  as  the  plug  is 
inserted. 

Plugs  should  not  be  rubbed  with  abrasives,  but  may  be 
cleaned  by  the  use  of  a  cloth  moistened  with  a  dilute  solution 
of  oxalic  acid.  The  resistance-box  tops  are  usually  made  of 
hard  rubber,  and  should  be  protected  from  dust,  moisture,  and 
prolonged  exposure  to  sunlight. 

The  resistance  of  a  clean  and  well-fitting  plug  contact  is 
under  0.0001  ohm.  With  all  the  plugs  in  place,  the  resistance 


FIG.  21. 


64        RESISTANCE  AND  ITS  MEASUREMENT     [II,  §  45 

across  the  top  of  an  average  resistance  box  will  be  between 
0.001  and  0.002  ohm. 

Resistance-box  coils  should  be  free  from  inductance  and 
capacity  as  nearly  as  possible.  The  former  may  be  largely 
eliminated  by  winding  the  wire  double,  so  that  the  magnetic 
fields  of  the  two  wires  may  annul  each  other.  Capacity  may 
be  effectively  reduced  by  reversing  the  direction  of  winding 
in  alternate  layers. 

46.  Rheostats.     The  name  rheostat  is  applied  to  a  device 
of  smaller  range  than  a  resistance  box,  but  with  a  larger  cur- 
rent-carrying capacity.     It  may  be  adjustable  continuously  or 
by  steps,  and  it  is  usually  not  important  to  know  the  actual 
resistance  values  which  correspond  to  the  various  settings.     It 
is  rated,  as  is  a  resistance  box,  in  terms  of  the  watts  dissi- 
pated without  overheating,  and  the  safe  maximum  current  is 
readily  inferred   from   the   temperature   of   the   coils,    which 
should  not  be   hot   to   the   touch.     Rheostats   are  commonly 
wound  with  manganin,  advance,  or  german  silver;  and  they 
may  t>e  wound  even  with  iron  wire  if  variations  in  resistance 
with  temperature  are  not  important.     Another  useful  form  is 
made  with  carbonized  cloth  in  disks,  or  with  carbon  blocks 
pressed  together  by  means  of  a  screw,  operated  by  a  hand 
wheel.     This  arrangement  affords  a  continuously  variable  re- 
sistance of  extreme  fineness  of  adjustment.     Tanks  of  liquid, 
such  as  water  solutions  of  copper  sulphate  or  common  salt, 
when  provided  with  suitable  electrodes,  make  useful  continu- 
ous rheostats. 

47.  Standard  Resistance  Coils.     Standard  coils  of  a  single 
fixed  value  are  used  for  precision  work,  and  are  usually  made 
in  the  form  shown  in  Fig.  22.     A  unit  of  this  type  consists  of 
a  coil  of  manganin  wire,  wound  non-inductively  on  a  brass 
tube,  carefully  insulated  from  it  and  covered  with  a  coating 
of  a  protective  varnish.     The  ends  are  attached  to  curved  ter- 


II,  §  47] 


GENERAL  DEFINITIONS 


65 


minals  arranged  to  dip  into  mercury  cups,  or  more  frequently 
they  are  provided  with  separate  current  and  potential  binding 
posts.  For  work  of  commercial  accuracy  these  coils  may  be 
used  in  air;  but  for  work  of  great  precision  they  are  im- 
mersed in  a  suitable  oil  bath,  and  are  provided  with  devices 


FIG.  22. 

for  controlling  the  temperature.  Such  standard  coils  of  any 
desired  degree  of  precision  up  to  one  or  two  parts  in  a  hun- 
dred thousand  may  be  purchased ;  their  range  extends  from 
100,000  ohms  to  0.0001  ohm,  or  even  to  0.00001  ohm.  These 
standards  vary  greatly  in  current-carrying  capacity,  depend- 
ing on  the  purpose  for  which  they  are  made.  Some  knowl- 
edge of  this  capacity  should  be  obtained  from  the  maker  in 
order  to  avoid  overheating. 


66        RESISTANCE  AND  ITS  MEASUREMENT     [II,  §  48 

48.  Measurement  of  Resistance.  Aside  from  the  direct 
measurement  of  resistance  by  means  of  an  ammeter,  a  volt- 
meter, and  Ohm's  law,  two  fundamental  methods  may  be  em- 
ployed. Methods  of  the  first  kind  are  called  comparison 
methods ;  in  these,  the  resistance  to  be  measured  is  compared 
with  some  previously  determined  standard. 

Methods  of  the  second  kind  are  called  absolute  methods. 
In  these,  the  resistance  is  determined  in  absolute  measure 
without  reference  to  any  existing  standards. 

The  first  method  is  the  one  commonly  used ;  absolute 
methods  will  not  be  considered  in  the  present  chapter.  The 
reference  standards  used  will  be  carefully  calibrated  resist- 
ance boxes,  or  single-valued  coils. 


II,  §49]    MEASUREMENT  OF  MEDIUM  RESISTANCES   67 


PAET  II.     MEASUREMENT  OF  RESISTANCES  OF 
MEDIUM  VALUES 

49.  The  Wheatstone  Bridge.    Of  the  comparison  methods 

for  measuring  resistance,  the  Wheatstone  bridge  is  more  used 

than   any   other,   especially    for 

values   of   medium   range.     The 

method  depends  upon   the  fact 

that  in  a  branched  circuit  (Fig. 

23)  the  potential  drop  from  a  to 

d  must  be  the  same  over  both 

branches.     It  then  follows  that 

for  any  point   b  chosen  on  the 

upper  branch  abd,  there  must  be 

a  corresponding  point  c  on  the  lower  branch  acd,  at  which  the 

potential  is  the  same.     There  is,  then,  no  difference  of  potential 

between  these  points,  and 
a  galvanometer  connected 
between  these  points  will 
indicate  no  deflection. 

The  circuit  may  take 
the  form  shown  in  Fig. 
24,  where  three  of  the 
resistances  are  known 
and  one  of  them,  jR_,  is 
to  be  determined.  The 
current  strength  in  each 
of  the  arms  is  represented 
by  iu  i2,  ?'3,  and  ixt  respect- 
ively. An  adjustment 

FlG  24.  of  the  resistance  values 

may  be  made  such  that 

the  galvanometer  shows  no  deflection.     When  this  has  been 

done,  the  potential  drop  along  ab  is  the  same  as  that  along 


68        RESISTANCE  AND  ITS  MEASUREMENT     [II,  §  40 

ac,  and  also  the  potential  drop  along  bd  is  the  same  as  that 
along  cd.     These  relations  may  be  written  in  the  form 

(10)  ilRl  =  i,Rz. 

(11)  i2R2  =  ixEx. 

Dividing  (10)  by  (11),  we  find 

/19\  hR\ 


Since  no  current  is  passing  through  the  galvanometer,  ^  =  i'2 
and  i8  =  ix.     Then  (12)  becomes 


or 

(14)  R,  =  R^' 

M1 

The  pairs  of  junction  points,  ad  and  be  (Fig.  24),  are  called 
the  conjugate  points  of  the  bridge  circuit.  In  general,  the  posi- 
tions of  battery  and  galvanometer  are  interchangeable.  How- 
ever, if  the  resistance  of  the  galvanometer  is  greater  than  that 
of  the  battery,  which  is  usually  the  case,  the  galvanometer 
should  be  connected  between  the  junction  points  of  the  high- 
est two  and  the  lowest  two  resistances. 

60.  The  Meter  Bridge.  In  one  common  form  of  the 
Wheatstone  bridge  the  arms  Rz  and  R±  are  replaced  by  a 
straight  homogeneous  wire  of  uniform  cross-section.  This 
form  is  called  the  slide-wire  bridge,  or  meter  bridge.  Such 
a  bridge  is  represented  in  Fig.  25.  It  consists  essentially  of 
a  wire  DH,  one  meter  long,  stretched  over  a  graduated  scale, 
and  with  its  ends  soldered  to  massive  brass  straps  SS'.  With 
a  known  resistance  at  Rz  and  the  resistance  to  be  measured 
at  Rx,  a  galvanometer  across  BE,  and  a  battery  across  AC,  we 
have  essentially  the  arrangement  shown  in  the  simple  diamond 
form,  Fig.  24.  A  point  can  now  be  found  on  the  wire,  E,  at 


II,  §  50]    MEASUREMENT  OF  MEDIUM  RESISTANCES   69 

which,  if  contact  is  made,  no  current  flows  through  the  gal- 
vanometer.    We  may  then  write 


(15) 


Since    for    a   uniform   wire    resistances    are   proportional    to 
lengths,  the  ratio  of  the  resistances  of  the  two  parts  of  the 


FIG.  25. 


wire  to  the  right  and  left  of  E  is  the  same  as  the  ratio  of  their 
lengths.  The  actual  values  of  E2  and  Rl  are  not  then  neces- 
sary and  (15)  may  be  written 


(16) 

or 

(17) 


= 
R3     A' 


Were  the  apparatus  and  method  free  from  error,  equation 
(17)  would  at  once  give  the  value  of  the  resistance  required. 
Certain  errors  are  inevitable,  however.  The  most  important 
of  these  are  the  following : 

(1)  The  tapping  edge  which  makes  contact  at  the  point  E, 
Fig.  25,  may  not  be  exactly  in  line  with  the  pointer  by  means 
of  which  the  position  is  read  on  the  scale. 


70        RESISTANCE  AND  ITS  MEASUREMENT     [II,  §  50 

(2)  In  order  to  form  the  Wheatstone  bridge  proportion  R3 
ought  to  include,  besides  the  resistance  of  the  coil  itself,  that 
of  the  connecting  wires  and  brass  plates  from  A  to  B,  and  also 
the  extra  resistances  introduced  at  the  points  where  connec- 
tions are  made.  Similarly  Rx  ought  to  include  the  correspond- 
ing resistances  from  B  to  (7;  RI  those  from  A  to  E,  and  R2 
those  from  C  to  E.  These  extra  resistances  are  usually  very 
small  in  the  case  of  R3  and  Rx,  but  for  Rl  and  R2  they  fre- 
quently are  not  negligible,  because  the  bridge  wire  may  not 
have  been  soldered  to  the  plates  exactly  at  the  ends  of  the 
scale. 

We  will  consider  a  method  for  avoiding  or  minimizing  these 
errors.  Suppose  that  the  tapping  edge  at  E  touches  the  wire 
at  a  point  which  is  d  cm.  nearer  H  than  the  scale  indicates, 
and  suppose  that  a  length  et  cm.  of  the  bridge  wire  would  have 
to  be  added  to  the  length  of  Lv  in  order  to  correct  for  extra 
resistances  from  A  to  D  and  at  D.  Further  suppose  that  a 
length  e2  cm.  of  the  bridge  wire  would  have  to  be  added  to  the 
length  of  L2  in  order  to  correct  for  the  extra  resistances  at 
that  end  of  the  bridge.  Then,  by  the  law  of  the  bridge 

ft  o\  RX  _  A  —  ^  4-  e2 

R3     Li  +  d  +  ei 

This  equation  contains  four  unknown  quantities,  and  it  is  prob- 
able that  some  of  them  can  be  eliminated  if  another  equation 
can  be  written.  This  can  be  done  if  we  interchange  Rz  and  Rx, 
and  obtain  another  balance  at  some  point  E',  probably  very 
close  to  E.  Let  DE'  be  called  L\  and  E'H  be  called  L'2. 
Then  the  law  of  the  bridge  gives 

/19>j  R^L'i  +  d  +  ei 

R3     L'2-d  +  ez 

From  the  law  of  composition  in  proportion  it  is  known  that  if 


II,  §  51]    MEASUREMENT  OF  MEDIUM  RESISTANCES    71 

then 

a  _  c  -\-  e 


Applying  this  law  to  (18)  and  (19),  we  have 

(20)      •Rx  =  L2  —  d  +  es  +  L'l  +  d  +  e1  =  L2  +  L\ 

' 


It  is  seen  that  the  quantity  d  no  longer  appears  in  the  equa- 
tion. Hence  by  taking  two  readings  with  Rx  and  R3  inter- 
changed, the  so-called  tapping  error  has  been  eliminated.  A 
further  simplification  of  the  formula  may  be  made  by  writing 
100  -  Z,!  for  Z2,  and  100  -  L\  for  L\.  Equation  (20)  then 
becomes 


The  quantities  el  and  e2  w^  he  small  as  compared  to  100  cm. 
if  the  bridge  has  been  carefully  made,  and  from  (21)  it  is  seen 
that  when  (Z^  —  L\)  is  small,  the  presence  of  the  term  (e^  +  e2) 
makes  very  little  difference  in  the  ratio  R£/RZ.  In  this  case 
ei  -f  e2  may  be  neglected  without  sensible  error,  and  (21) 
becomes 

flj  =  100  -(A  -L\) 

Rs     100+  (A  -L\) 

In  order  to  make  (L±  —  L\)  small,  it  is  only  necessary  to 
choose  RZ  near  Rx  in  value,  so  that  the  contact  point  E  is  near 
the  middle  of  the  scale. 

51.  Laboratory  Exercise  VI.  To  measure  a  resistance  with 
the  meter  bridge. 

APPARATUS.  Meter  bridge,  portable  or  other  sensitive  gal- 
vanometer, tap  key,  one  or  two  dry  cells,  adjustable  resistance 
box,  and  the  samples  to  be  measured. 

PROCEDURE.  (1)  Arrange  the  circuit  as  in  Fig.  25.  Select 
some  value  of  R3  which  you  judge  to  be  of  the  same  order  of 


72         RESISTANCE  AND  ITS  MEASUREMENT     [II,  §  51 

magnitude  as  the  resistance  to  be  measured.  Tap  the  sliding 
contact  lightly  at  opposite  ends  of  the  wire  and  note  whether 
there  is  a  reversal  of  the  galvanometer  deflection.  If  no 
reversal  occurs,  there  is  some  fault  in  the  circuit  which  must 
be  sought  for  and  corrected.  If  a  reversal  does  occur,  note 
the  scale  reading  and  calculate  roughly  the  value  of  Rx  to  the 
nearest  ohm.  Set  7?3  at  this  value,  and  again  seek  the  point 
at  which  reversal  occurs.  This  should  now  be  near  the  middle 
of  the  wire. 

(2)  Having  found  this  position  E,  let  the  notebook  sketch 
show  the  relative  positions  of  j£3,  Rx,  Ll}  and  Z2,  and  record 
the  values  of  Rz  and  L^     Without  altering  the  value  of  R3, 
interchange  Rz  and  Rx  and  again  locate  the  balance  point. 
Record  the  new  value  of  L\.     Also  note  and  record  the  dis- 
tance through  which  the  slider  can  be  moved  from  the  balance 
point  before  the  least  observable  deflection  occurs  on  the  gal- 
vanometer, and  state  the  probable  precision  of  the  settings. 

Repeat  each  reading  several  times,  making  independent 
settings  each  time.  The  balance  point  should  be  located  with 
the  attention  on  the  galvanometer  and  not  on  the  scale,  in 
order  to  avoid  being  influenced  by  the  previous  setting. 

(3)  To  obtain  the  value  of  Rt  substitute  in  equation  (22). 

It  may  be  observed  that  the  galvanometer  deflects  when  the  tapping 
contact  is  pressed,  even  though  the  battery  key  is  open.  This  means 
that  thermal  electromotive  forces  are  acting  at  some  of  the  contacts. 
The  average  of  readings  taken  with  the  terminals  of  the  battery  reversed 
will  be  free  from  errors  due  to  this  cause.  The  battery  key  should  in- 
variably be  pressed  before  the  key  in  the  galvanometer  circuit,  in  order 
to  insure  the  current  rising  to  its  full  value  before  the  galvanometer  is 
connected.  Otherwise  there  may  be  a  slight  deflection  due  to  induction' 
in  some  part  of  the  circuit.  The  keys  should  be  kept  closed  only  as  long 
as  may  be  necessary  to  take  readings,  in  order  to  prevent  any  heating 
effects. 

62.  The  Box  Bridge.  Another  common  form  of  the  Wheat- 
stone  bridge  is  the  box  bridge,  or  post-office  bridge,  so-called 


II,  §  52]    MEASUREMENT  OF  MEDIUM  RESISTANCES    73 

because  it  was  used  at  an  early  date  by  the  British  Post  and 
Telegraph  Office.  A  top  view  of  the  circuit  of  one  form 
of  this  bridge  is 
shown  in  Fig.  26. 


i 


FIG.  26. 


The  resistance  to 
be  measured  is 
connected  at  723; 
RI  and  724  are 
called  the  ratio 
arms,  and  R2  is 
called  the  rheostat 
arm.  A  little 
study  of  the  cir- 
cuit will  show  that 
when  the  ratio  of  Rl  to  R4  is  unity,  the  bridge  will  be  balanced 
when  R2  is  equal  to  R3.  In  any  case,  when  the  resistances 
have  been  adjusted  so  that  there  is  no  galvanometer  deflec- 
tion, we  have 

(23)  KI  =  B?. 

If  RI  =  RU  obviously  R2  =  R^.  However,  if  R±  equals  1/10, 
or  1/100,  or  1/1000,  of  7?4,  then  R3  must  be  respectively  10, 
100,  or  1000  times  R2  in  order  that  a  balance  may  exist.  With 
the  highest  value  in  R2  equal  to  11,110  ohms,  and  the  ratio 
R^R4  equal  to  1/1000,  a  resistance  in  R3  as  high  as  11,110,000 
ohms  can  be  measured.  In  case  the  ratio  arms  can  be  reversed, 
the  bridge  equation  becomes 


= 
RI    R,' 

and  if  R4  is  1000  and  Rl  is  10,  then  for  a  balance  R2  will  be 
100  times  R3.  The  smallest  value  in  Rz,  which  is  usually  one 
ohm,  is  then  equivalent  to  0.01  ohm  in  R^.  If  the  highest  ratio 


74        RESISTANCE  AND  ITS  MEASUREMENT     [II,  §  52 

of  RI  to  jRi  is  1000  to  1,  the  smallest  fraction  measurable  in 
Rz  with,  one  ohm  as  the  least  step  in  R2)  is  0.001  ohm. 

Box  bridges  commonly  have  ratio  arms  with  a  ratio  of  1000 
to  1,  and  rheostat  arms  with  a  range  from  1  to  1000,  or  higher. 
This  gives  a  very  large  theoretical  range,  but  practically  the 
method  is  limited,  and  its  greatest  value  is  in  the  middle  range 
from  one  ohm  to  perhaps  a  megohm.  For  higher  values  than 
this  an  increased  potential  difference  is  necessary,  and  insu- 
lation trouble's  appear.  Moreover,  a  high  degree  of  precision 
is  required  in  the  ratio  coils.  For  high  resistances,  however, 
advantage  may  be  taken  of  the  law  of  parallel  circuits,  and  a 
known  resistance  may  be  connected  in  parallel  with  the  un- 
known high  resistance,  the  combined  value  of  the  pair  being 
thus  made  low  enough  to  fall  well  within  the  precision  limits 
of  the  bridge. 

For  low  values  of  less  than  one  ohm,  the  method  gives  less 
accurate  results  because  of  the  errors  due  to  contact  resistance. 

Detailed  descriptions  of  the  great  variety  of  box  bridges  now 
in  use  are  not  practicable  in  a  textbook.  When  confronted 
with  an  unfamiliar  type,  the  student  should  first  make  a 
sketch  of  the  simple  diamond  form  of  the  bridge  circuit,  and 
then  one  of  the  actual  connections  of  the  box  bridge  in  hand. 
In  this  way  it  is  easy  to  identify  the  ratio  arms,  rheostat  arm, 
and  the  unknown  resistance,  together  with  the  proper  points 
of  connection  for  the  battery  and  the  galvanometer. 

53.  Laboratory  Exercise  VII.  To  measure  a  resistance  with 
the  box  bridge. 

APPARATUS.  Box  bridge,  portable  or  other  sensitive  galva- 
nometer, one  or  two  dry  cells,  and  the  samples  to  be  measured. 

PROCEDURE.  (1)  Connect  the  resistance  to  be  measured  to 
the  line  terminals,  and  the  battery  and  galvanometer  to  their 
respective  terminals.  Make  the  ratio  arms  equal.  With  the 
rheostat  arm  equal  to  zero,  press  the  battery  key  firmly,  then 


II,  §54]    MEASUREMENT  OF  MEDIUM  RESISTANCES    75 

cautiously  and  lightly  tap  the  galvanometer  key  and  note 
the  direction  of  the  galvanometer  deflection.  Then  make  the 
rheostat  arm  as  large  as  possible,  or  withdraw  the  infinity 
plug  if  there  is  one,  and  again  tap  the  keys  as  before  and  note 
the  direction  of  the  deflection.  If  the  circuit  has  been  cor- 
rectly set  up,  the  two  deflections  will  be  in  opposite  directions. 
If  this  is  not  the  case,  the  fault  must  be  found  and  remedied, 
as  it  is  useless  to  attempt  to  secure  a  balance  unless  the  de- 
flection reverses  in  the  above  test. 

It  is  not  necessary  to  wait  for  the  galvanometer  to  come  to 
rest  except  during  the  final  adjustment.  The  galvanometer 
key  should  be  tapped  with  a  quick  motion,  so  that  a  lack  of 
balance  is  indicated  without  the  risk  of  a  too  violent  fling  of 
the  suspended  system. 

(2)  Having  found  the  approximate  resistance  with  an  even 
ratio,  step  at  once  to  the  highest  ratio  which  the  box  will  allow, 
or  which  the  measurement  requires,  and  find  the  value  of  the 
rheostat  arm  necessary  for  a  balance.  Find  the  value  of  the 
unknown  resistance  and  confirm  this  by  several  repetitions,  also 
with  different  settings  of  the  ratio  arms. 

QUESTIONS.  (1)  What  will  be  the  effect  of  using  more  or 
less  battery  cells  ? 

(2)  What  governs  the  choice  of  values  in  the  ratio  arms  ? 

(3)  How  would  you  adjust  the  ratio  arms  in  order  to  meas- 
ure (a)  a  resistance  higher  than  the  range  of  the  rheostat  arm  ? 
(6)  a  resistance  lower  than  the  smallest  step  in  the  rheostat  arm  ? 

54.  Laboratory  Exercise  VIII.  To  verify  the  laws  of  series 
and  parallel  resistance  combinations. 

APPARATUS.  Three  or  more  samples  to  be  measured  and 
box  bridge  with  accessories. 

PROCEDURE.  (1)  Measure  in  the  prescribed  way  the  value  of 
the  resistances  singly  and  then  in  various  series  and  parallel 
groups. 


76         RESISTANCE  AND  ITS  MEASUREMENT     [II,  §  54 

(2)  Calculate  the  equivalent   resistance   for   each  of   these 
groups,  and  compare   the   result  with  the  value   found  from 
measurement.     Express  the  percentage  deviation  of  the  single 
values  and  show  the  effect  of  these  deviations  on  the  computed 
results. 

(3)  Tabulate  all  data  and  results.     Prove  in  full  the  formu- 
las used. 


55.  Galvanometer  Resistance.  A  precise  determination  of 
the  resistance  of  a  galvanometer  is  best  made  by  means  of  the 
box  bridge.  Connect  the  galvanometer  to  the  line  terminals 
of  the  bridge,  taking  care  that  the  suspended  system  is 
arrested,  or  supported  in  such  a  manner  that  it  will  not  be 
deflected  by  the  current  which  passes  through  it. 

Frequently,  however,  an  approximate  value  will  suffice,  in 
which  case  some  modification  of  the  half-deflection  method  is 
convenient.      Two   methods    of   pro- 
cedure will  be  described. 

CASE  I.  For  finding  the  approxi- 
mate resistance  of  a  galvanometer 
which  has  a  straight-line  calibration 
curve,  connect  as  in  Fig.  27,  using  a 
gravity  battery  because  of  its  freedom 
from  polarization.  Fix  R  at  some  value  Slt  such  that  the 
galvanometer  gives  a  full-scale  deflection,  the  battery  being 
shunted,  if  necessary,  with  a  low  resistance.  Ohm's  law  for 
the  entire  circuit  may  be  written  in  the  form 


FIG.  27. 


(24) 


Fdl  = 


E 


BS 

B  +  S 


If  the  resistance  R  is  now  increased  to  some  value  R^  such 
that  the  galvanometer  deflection  ds  is  equal  to  one  half  of  dly 
then 


II,  §55]    MEASUREMENT  OF  MEDIUM  RESISTANCES    77 

(25)      j'=jiH2+g+^' 

Dividing  (25)  by  (26),  we  find 


2  = 


r 
(26) 


If  B  is  small  as  compared  to  g,  it  may  be  neglected.  More- 
over, if  8  is  low  enough  so  that  R^  may  be  made  equal  to 
zero,  then,  by  (26),  g  =  R2. 

CASE  II.  When  the  battery  to  be  used  is  one  that  polarizes 
rapidly,  the  foregoing  method  is  useless,  as  the  low  resistance 
shunt  would  permit  the  cell  to  run 
down.  In  this  case  arrange  a 
circuit  as  in  Fig.  28,  in  which  the 
battery  is  in  series  with  a  high 
resistance  AC,  and  let  the  poten- 
tial difference  between  A  and  P 
be  that  used  in  the  galvanometer 
circuit.  Let  V  represent  the 
potential  difference  between  A 

and  P,  and  let  r  be  the  resistance  of  this  portion  of  the  circuit. 
The  position  of  P  may  be  so  chosen  that  a  full-scale  deflection 
results  with  R  =  0.  The  current  through  the  galvanometer  is 
then  given  by  the  formula 

(27)       '  *.-**-;£• 

If  R  is  now  increased  to  some  value  ^ ,  such  that  the  galva- 
nometer deflection  is  reduced  to  one  half  its  former  value,  we 
have 


78        RESISTANCE  AND  ITS  MEASUREMENT     [II,  §  55 

/OQ\  T  TT^^'l 

I*  =  F2=r-.,-g  +  It1' 
Dividing  (27)  by  (28),  and  assuming  that  V  remains  constant,1 

2  =  r  +  g  +  R,t 

or 

(29)  g  =  Ri~  r. 

56.  Battery  Resistance.  The  internal  resistance  of  a  bat- 
tery will  be  more  fully  discussed  in  Chapter  III.  Often  for 
a  battery  of  fairly  high  resistance  and  constant  electromotive 
force,  such  as  the  gravity  battery,  a  method  similar  to  that 
described  in  the  preceding  article  will  be  convenient.  Connect 
as  in  Fig.  27,  but  with  the  shunt  across  the  galvanometer 
instead  of  across  the  battery.  Make  R  some  value,  say  Rl> 
such  that  the  deflection  d^  is  about  the  full  scale.  Increase  R 
to  some  value  R^  such  that  the  corresponding  deflection  d2  is 
just  one  half  of  dlt  Then,  by  reasoning  similar  to  that  used 
in  deriving  equation  (26)  it  can  be  proved  that 

/um                                                   7?         7?          9    T?  ^9 

^OUJ  _O  =  Xt2  —  ^  •"! 


If  the  galvanometer  is  not  shunted,  and  if  Rl  can  be  made 
zero,  we  have 

(31)  B  =  R2-g. 

In  case  g  is  shunted  with  a  very  low  value  so  that  sg/(s  -f-  g)  is 
negligible,  and  if  R±  =  0,  we  have 

(32)  B  =  M2. 

1  Changes  in  R  will  change  the  value  of  V.  The  student  should  work  out 
this  relation  carefully  for  any  particular  case  in  which  the  method  is  used. 
In  general,  the  method  will  be  used  only  when  approximate  results  for  g  will 


II,  §  58]     MEASUREMENT  OF  LOW  RESISTANCES        79 


PART  III.     METHODS  FOR  MEASURING  Low 
RESISTANCES 

57.  Inadequacy  of  the  Wheatstone  Bridge.     Although  the 
various  forms  of  the  Wheatstone  bridge  are  capable  of  great 
precision  when  applied  to  the  measurement  of  resistances  of 
medium  values,  they  are  not  suited  for  small  resistances  of 
the  order  of  fractions  of  an  ohm  because  of  contact  errors. 
Hence  other  methods  are  required  which  are  free  from  these 
errors,  and  which  will  enable  a  resistance  of  the  order  of  0.001 
ohm  to  be  measured  with  the  same  precision  as  that  attained 
with  a  Wheatstone  bridge  on  a  1000-ohm   coil.     By  far  the 
most  widely  used  method  for  measuring  low  resistances  is  the 
one  devised  by  Lord  Kelvin,  and  known  as  the  Thomson  or 
Kelvin  bridge.     Its  development  may  be  traced  through  four 
stages,  and  will  be  considered  in  the  four  articles  that  follow. 

58.  The  Kirchhoff  Bridge.     The  circuit  shown  in  Fig.  29 
represents  a  device  which  is  not  now  in  use,  but  which  con- 
tains     the      funda- 
mental    idea    from 

which     present-day  ' 

methods  have  arisen. 
It  depends  upon  the 
use  of  a  differential 
galvanometer,  which 
is  simply  a  double 
instrument,  symmetrically  arranged,  so  that  equal  currents 
through  the  two  coils  will  leave  the  equilibrium  of  the  sus- 
pended system  unchanged.  If  such  a  galvanometer  has  its 
respective  pairs  of  terminals  connected  across  the  standard 
resistance  s  and  the  unknown  resistance  x,  and  if  a  constant 
current  is  maintained  through  s  and  x  in  series  by  the  battery 
B,  there  will  be  no  deflection  when  the  two  resistances  are 
equal. 


FIG.  29. 


80        RESISTANCE  AND  ITS  MEASUREMENT     [II,  §  59 


fV  J 

\    r) 

D 

N 

—S^    .^     s<.     X\. 

F 

0-J  0—  j 

x\    .*_ 

X 

R 

\ 

^^s 

il 

59.  A  Direct-deflection  Method.  In  this  method  the  two 
resistances  may  be  compared  by  means  of  any  sensitive  galva- 
nometer, arranged  as  in  Fig.  30.  This  galvanometer  is  con- 
nected to  the  middle  points  of  a  double-pole,  double-throw 
/--^  switch  Z>,  and  can 

be  successively  con- 
nected across  the 
terminals  of  s  or  x. 
The  corresponding 
deflections  will  be 
shown  to  be  propor- 
tional to  the  resist- 

r  IG.  C$0. 

ances  respectively. 

Let  Va  and  Vx  be  the  potential  differences  respectively  be- 
tween the  terminals  NN  of  the  standard  resistance  s,  and  the 
terminals  XX  of  the  unknown  resistance  x,  when  a  suitable 
current  is  maintained  through  them  in  series  by  the  battery 
B.  When  the  switch  connects  the  galvanometer  across  the 
terminals  of  s,  the  current  through  the  galvanometer  is  given 
by  the  equation 

(33)  t!=^. 

Similarly,  when  the  switch  connects  the  galvanometer  across 
the  terminals  of  xy 

(34)  i2  =  ^- 

Dividing  (33)  by  (34),  we  have 

(35)  i  =  £.' 
But 

(36)  $  =  ^; 
hence  it  follows  that 

/OfTX  rf          di 

(37)  17  =7  * 

V,     dz 


II,  §  59]     MEASUREMENT  OF  LOW  RESISTANCES       81 

If  the  currents  through  s  and  x  are  represented  by  i.  and  ix, 
then,  by  Ohm's  law,  we  have 

(38)  «.  =  £, 
and 

(39)  i,^, 

If  the  galvanometer  resistance  is  relatively  high,  so  that  an 
inappreciable  current  is  diverted  from  the  main  circuit  when 
it  is  placed  in  parallel  with  s  or  #,  then  is  may  be  considered 
equal  to  ix.  Hence,  equating  (38)  and  (39),  we  find 


Sam 
JU 

or 

(40)  iH- 

Combining  (40)  and  (37),  we  have 

whence 

(41)  x  =  s^. 

The  unknown  resistance  is  then  readily  calculated  when  s 
is  a  standard  of  known  value.  A  high  value  of  the  galvanom- 
eter resistance  becomes  of  less  importance  as  the  values  of  s 
and  x  become  more  nearly  equal.  When  they  are  equal,  the 
same  current  is  diverted  through  the  galvanometer  in  each 
case,  and  i8  is  then  rigorously  equal  to  ix.  It  is  important, 
then,  to  select  a  standard  of  as  nearly  as  possible  the  same 
value  as  the  sample  to  be  measured.  In  such  measurements 
large  errors  may  enter,  unless  changes  in  resistance  with  tem- 
perature are  taken  into  account. 
o 


82       RESISTANCE  AND   ITS  MEASUREMENT     [II,  §  60 

60.  A  Zero  Method.  The  arrangement  shown  in  Fig.  31 
may  be  used  to  compare  an  unknown  low  resistance  with 
a  standard.  This  method  assumes  a  continuously  adjustable 


Fia.  31. 

standard  resistance,  which  is  difficult  to  realize  in  practice. 
It  depends  upon  the  equalization  of  the  potential  drops  on  the 
two  sides  of  the  galvanometer,  in  which  case  the  deflection  is 
zero. 

61.  The  Double  Bridge.  As  a  direct  development  of  the 
method  described  in  §  60,  four  resistances  are  added  to  the 
circuit,  as  shown  in  Fig.  32,  and  in  this  form  it  is  usually 
known  as  the  Kelvin  bridge  or  the  Thomson  bridge. 


FIG.  32. 


Let  s  represent  a  known  standard  resistance,  and  x  the 
resistance  of  the  sample  to  be  measured.  The  resistance  coils 
A,  B,  a,  and  b  are  fixed  in  value  so  that 


(42) 


B~b 


II,  §  61]     MEASUREMENT  OF  LOW  RESISTANCES       83 

It  will  be  proved  below  that  when  the  various  resistances  are 
adjusted  so  that  the  galvanometer  shows  no  deflection,  we 
have  the  relation 


When  the  resistances  in  the  bridge  have  been  so  arranged 
that  no  current  passes  through  the  galvanometer,  the  current 
ii  in  A  equals  that  in  B.  Similarly,  the  current  iz  in  a  equals 
that  in  b,  and  the  current  i3  in  s  equals  that  in  x.  Then, 

(43)  si*3  +  ai2  =  Aii. 

(44)  xis  +  biz  =  Bi^ 

Dividing  both  equations  by  their  right-hand  members,  we  have 

(45)  ff  +  ^r2=L 
Aii     Aii 

(46)  **.+  ***=!. 

Bh   ml 

From  (42)  a/  A  =  b/B  ;  hence  a  j  A  may  be  substituted  in  (46) 
for  b/B.  Then,  subtracting  (46)  from  (45),  we  find 

(47)  ^-^  =  0, 
Aii     Bi± 

or- 

s  _  x 

A~B' 
whence 

(48)  1-5- 

which  is  the  relation  desired.  The  values  of  A  and  B  may  be 
made  very  large,  and  in  comparison  with  them  the  resistances 
of  the  lead  wires  and  contacts  may  be  neglected. 

The  Kelvin  bridge  may  be  built  up  from  resistance  boxes 
available  in  the  laboratory.  However,  the  usefulness  of  the 


84       RESISTANCE  AND   ITS  MEASUREMENT     [II,  §  61 

method  has  led  to  the  development  of  a  convenient  and  self- 
contained  box  of  coils,  designed  to  yield  results  with  a  mini- 
mum of  time  and  labor.  Such  a  set  is  shown  in  Fig.  33  and 


FIG.  33. 

includes  in  a  single  case  all  those  parts  of  the  circuit  in 
Fig.  32  which  are  above  NN  and  XX,  exclusive  of  the 
galvanometer. 

In  the  group  of  coils  under  switch  x  100,  there  are  twenty 
coils  of  100  ohms  each,  ten  in  each  semicircle.     In  the  group 


II,  §  61]     MEASUREMENT  OF  LOW  RESISTANCES       85 

marked  x  10,  there  are  eighteen  coils  of  10  ohms  each,  nine  in 
each  semicircle.  In  the  group  marked  x  1  there  are  eighteen 
1-ohm  coils,  and  in  the  group  marked  x  0.1  there  are  eighteen 
0.1-ohm  coils.  In  each  case  there  are  the  same  number  of 
coils  in  the  upper  as  in  the  lower  semicircles.  The  contact 
brushes  of  the  rotating  switches  over  the  semicircles  in  each 
group  are  insulated  from  one  another.  It  is  readily  seen 
that  as  any  switch  is  rotated,  the  resistance  effective,  in  the 
upper  and  lower  semicircles  of  any  dial  group  is  always  the 
same. 

In  measuring  a  resistance  with  this  apparatus,  a  working 
battery  WB,  Fig.  32,  is  put  in  series  with  a  properly  chosen 
standard  resistance  s,  and  the  resistance  to  be  measured  x.  A 
control  rheostat  should  be  included  in  this  circuit.  The  termi- 
nals of  the  standard  resistance  are  connected  to  the  binding 
posts  NNy  and  those  of  the  resistance  to  be  measured  to  XX. 

The  effective  resistance  of  all  four  of  the  upper  semicircles 
taken  in  series  corresponds  to  B  of  Fig.  32,  while  that  of  the 
four  lower  semicircles  corresponds  to  b.  The  group  of  three 
plug  controlled  resistances  at  the  left  in  Fig.  33  corresponds 
to  A,  and  that  near  the  middle  of  the  diagram  corresponds  to 
a.  The  rotation  of  any  one  of  the  four  dial  switches  to  the 
right  or  to  the  left  will  increase  or  decrease,  respectively,  the 
values  of  B  and  b  by  equal  amounts.  Since  A  and  a  will 
always  be  equal,  it  is  seen  that  the  ratio  A/B  will  always  be 
equal  to  a/b,  which  is  the  fundamental  condition  upon  which 
the  operation  of  the  bridge  is  based.  Inasmuch  as  this  is  a 
zero  method,  its  operation  is  not  affected  by  fluctuations  in 
the  working-battery  current. 

The  method  is  widely  used  in  the  comparison  of  low  resist- 
ances. It  is  available  also  for  calibrating  standard  coils,  for 
measuring  the  variation  of  resistance  with  changes  in  tempera- 
ture, and  for  finding  the  conductivity  of  bars  and  wires  in  the 
form  of  short  samples. 


86          RESISTANCE  AND   ITS  MEASUREMENT     [II,  §62 

62.  Laboratory  Exercise  IX.  To  measure  a  low  resistance 
with  the  Kelvin  bridge,  and  to  find  the  temperature  coefficient  of 
resistance  for  a  sample  of  wire. 

APPARATUS.  Standard  resistance  of  low  value,  samples  to 
be  measured,  oil  bath  with  heater,  Kelvin  bridge,  galvanometer, 
a  few  cells  of  storage  battery,  tap  key,  and  connecting 
wires. 

PROCEDURE.  (1)  Connect  in  series  with  the  battery  and  a 
control  rheostat  the  standard  resistance  s,  and  the  sample  to 
be  measured  x.  Connect  the  terminals  of  s  to  the  points 
marked  NN  on  the  bridge  and  the  terminals  of  x  to  the  points 
XX.  Include  a  tap  key  k  in  series  with  the  galvanometer, 
which  is  connected  at  G,  Fig.  33. 

(2)  Introduce  equal  resistances  at  A  and  a,  and  with  all  the 
dial  switches  on  zero,  tap  k  and  note  the  direction  of  the  throw. 
Set  the  switch  x  100  on  10  and  again  tap  k.     If  the  galvanom- 
eter does  not  deflect  in  the  opposite  direction,  it  is  necessary 
to  interchange  the  wires  at  either  NN  or  XX.     If  reversal 
occurs,  a  setting  of  the  four  dials  may  be  found  such  that  there 
is  no  deflection  when  k  is  closed. 

(3)  Using  the  notation  of  §  61,  the  upper  halves  of  all  the 
dials  will  read  the  value  of  B,  and  the  lower  halves  of  the 
dials  will  read  the  value  of  b,  which  values  in  this  apparatus 
will  always  be  the  same.     Calculate  the  value  of  x  from  equa- 
tion (48). 

(4)  With  the  sample  immersed  in  oil  or  water,  make  sev- 
eral measurements  at  carefully  determined  temperatures,  and 
calculate  the  value  of  the  temperature  coefficient  from  equation 
(9),  §  42.     The  bath  must  be  stirred  so  that  the  temperature 
is  uniform  throughout. 

63.  Laboratory  Exercise  X.  To  measure  a  low  resistance 
by  the  fall  of  potential,  direct-defiection  method,  and  to  find  the 
resistivity  of  a  sample  of  wire. 


II,  §  64]     MEASUREMENT  OF  LOW  RESISTANCES      87 

APPARATUS.  One  or  two  low-resistance  standards,  samples 
to  be  measured,  double-pole  double-throw  switch,  reversing 
switch,  galvanometer,  one  or  two  constant  battery  cells,  resist- 
ance box,  and  clamps. 

PROCEDURE.  (1)  Fasten  the  clamps  to  the  edge  of  the  table 
and  secure  the  ends  of  a  sample  of  wire  15  to  30  cm.  long  in 
the  clamps,  stretching  it  tight.  Arrange  the  circuit  as  in 
Fig.  30,  and  include  a  reversing  switch  between  the  galvanom- 
eter and  the  switch  D. 

Connect  current  and  potential  wires  to  the  double  binding 
posts  on  the  clamps,  and  adjust  R  until  suitable  deflections 
are  observed.  Connections  should  be  so  made  that  galvanom- 
eter deflections  are  on  the  same  side  of  the  scale  when  the 
switch  D  is  thrown  over. 

(2)  Take  reversed  galvanometer  deflections  across  x  and  s, 
repeating  several  times  for  each  sample. 

(3)  Calculate  from  equation  (41)  the  value  of  the  resistance 
of  the  sample. 

(4)  Measure  the  length  and  diameter  of  the  wire  used  and 
calculate  the  resistivity  in  microhms  per  centimeter  cube,  for 
each  sample  furnished,  using  equation  (3),  §  39. 

Note  the  resistance  of  the  galvanometer  used.  Is  it  high  enough  to 
avoid  sensible  error  ? 

Discuss  the  effect  on  the  results  obtained  if  the  galvanometer  resistance 
were  increased  or  decreased.  State  the  probable  precision  attained  in 
the  apparatus  used. 

For  many  commercial  purposes  a  moderate  degree  of  precision  suffices 
and  a  millivoltmeter  may  be  used  instead  of  the  galvanometer.  The  ratio 
of  the  readings  will  give  the  ratio  of  the  resistances. 

64.  Measurement  of  Low  Resistances  by  Ohm's  Law.    A 

simpler  method  than  the  foregoing,  and  one  which  is  suffi- 
ciently accurate  for  a  great  deal  of  routine  testing,  depends 
on  Ohm's  law,  and  upon  the  direct  measurement  of  current 
and  potential  difference  with  ammeter  and  millivoltmeter. 


88        RESISTANCE  AND  ITS  MEASUREMENT     [II,  §  64 

Consider  a  circuit  arranged  as  in  Fig.  34,  in  which  x  is  the 
resistance  to  be  measured.     Let  /  and  V  represent  the  read- 
^^  ings    of    the    ammeter    and 

millivoltmeter,  respectively, 
and  let  r  be  the  resistance  of 
the  millivoltmeter.  Writing 
Ohm's  law  for  the  portion  of 
the  circuit  between  a  and  b, 


<                      Y                    -x 

a                                                   b        \ 
R                 G 

FIG.  34. 


we  have 


V 


xr 

x  -}-  r 

from  which  the  value  of  x  is  found  to  be 

V 
(49)  -T~V' 


If  r  is  relatively  large,  the  term  V/r  may  be  neglected,  and 
a  direct  application  of  Ohm's  law  gives  the  value  of  the  un- 
known resistance  x. 

65.  Laboratory  Exercise  XI.  To  measure  a  low  resistance 
with  ammeter  and  millivoltmeter. 

APPARATUS.  Ammeter,  millivoltmeter,  one  or  two  storage 
cells,  control  rheostat,  and  samples  to  be  measured. 

PROCEDURE.  (1)  Arrange  the  circuit  as  in  Fig.  34,  with  the 
contact  points  a  and  b  at  the  terminals  of  the  sample  to  be 
tested.  Adjust  R  so  that  the  current  and  potential  difference 
are  easily  readable  on  the  instruments. 

(2)  Compute  the  value  of  x  for  several  pairs  of  values  of  / 
and  V,  from  equation  (49),  both  with  and  without  the  factor 
V/r,  and  show  the  percentage  accuracy  attained. 


II,  §  67]     MEASUREMENT  OF  HIGH  RESISTANCE       89 

PART  IV.     MEASUREMENT  OF  HIGH  RESISTANCE 

66.  High-resistance  Methods.    It  has  been  pointed  out 
in  §  52  that  the  Wheatstone  bridge  is  not  available  for  the 
measurement  of  high  resistance.     The  procedure  most  com- 
monly employed  for  such  measurements  involves  the  direct 
observation  of  deflections.     Hence  it  is  not  a  zero  method. 
Moreover,  since   the   high   resistances   measured  are  usually 
those  of  insulation,  great  precision  is  not  sought. 

67.  A  Direct-deflection  Method.     The  following  is  a  rapid 
and  fairly  accurate  method  for  finding  the  value  of  a  high  re- 
sistance.     It   requires  a  galvanom- 
eter of   known  figure    of    merit  or 

a  sensitive  milliammeter,  a  volt- 
meter, a  sufficiently  high  electro- 
motive force,  and  the  sample  to  be  I — • — |  In  H  |- 
tested.  The  circuit  is  arranged  as 
in  Fig.  35.  The  galvanometer  be- 
comes an  ammeter  when  its  figure 
of  merit  is  known  and  may  then  FIG<  35. 

be    used    to    measure    the    current 
passing  through  the  circuit.     By  Ohm's  law,  we  have 

(50)  i  =  Fd  =  -¥~, 

where  F  is  the  figure  of  merit  of  the  galvanometer,  g.  is  its 
resistance,  V  is  the  terminal  potential  difference  of  the  bat- 
tery, and  a;  is  the  resistance  to  be  measured.  From  (50)  we 
find 

(51)  X=¥-g. 


If  g  is  small  compared  with  x,  it  may  be  dropped  for  approxi- 
mate results. 


90        RESISTANCE  AND   ITS  MEASUREMENT     [II,  §  68 

68.   A  Voltmeter  Method.     A  high  resistance  can  be  meas- 
ured also  by  means  of  a  circuit  arranged  as  shown  in  Fig.  36. 

A  battery  of  100  volts  or  up- 


wards   is   connected    in   series 

with    the    high    resistance    x. 

The  voltmeter  is  first  coii- 
^~^Fl  3-  nected  across  the  battery  ter- 

minals CA,  and  its  reading  V 

is  observed.  The  contact  point  is  then  changed  from  A  to 
B,  and  again  the  reading  of  the  voltmeter  V  is  observed. 
Writing  Ohm's  law  for  the  two  cases,  we  have 

(52)  1,  =  ^  =  ^, 

*7 

and 

(53)  i2=*V!2  =  -JL, 

g  +  x 

where  F  is  the  figure  of  merit  of  the  galvanometer,  g  is  its 
resistance,  and  E  is  the  electromotive  force  of  the  battery. 
The  internal  resistance  of  the  battery  is  negligible  in  compari- 
son with  the  high  resistance  in  series  with  it.  Dividing  (52) 
by  (53),  we  find 


(54)  i 

<*2       g 

Since  deflections  on  the  voltmeter  are  proportional  to  poten- 
tial differences,  we  have 

(55)     -  % 

or 

(56)  a 

This  method  will  not  yield  results  of  high  precision,  but  it 
is  convenient  and  requires  only  simple  equipment.  It  is  much 
used  in  practice  for  rapid  and  approximate  determinations. 


II,  § 


MEASUREMENT  OF  HIGH   RESISTANCE       91 


It  is  applied,  for  example,  in  the  case  of  finding  the  insulation 
resistance  of  a  wiring  circuit,  as  shown  in  Fig.  37.     The  dis- 
tributed   resistances    be- 
tween  the   two  sides  of 
the  line  and  the  ground 
G    are     represented    by 
a?!   and    #2>   respectively. 
With  the  instruments  as 
shown,   the   value   of   xl 

may  be  found.     In  order  FlG  37 

to  find   #2,   the   point   a 
must  be  changed  to  the  other  side  of  the  line. 

69.  The  Method  of  Substitution.  As  a  general  method  of 
good  precision  for  measuring  the  insulation  resistance  of  wires 
and  cables,  or  other  high  resistances,  the  following  method  is 

frequently  used.  With  a 
circuit  arranged  as  in  Fig. 
38,  let  R±  be  some  known 
resistance  of  one  tenth 
megohm  or  higher,  and  let 
Rz  be  the  unknown.  The 
galvanometer  must  be  one 
of  high  current  sensibility, 
and  the  battery  should  be 

capable  of  supplying  a  constant  potential  difference  of  from 
one  hundred  to  five  hundred  volts.  With  the  switch  K  raised 
to  the  point  u,  the  current  through  the  galvanometer  is  given 
by  the  equation 

/K7\  -•  TTT-J  E 

{o  t  j  i\  =  Jc  1*1  ^  j 

•Kl  ~\~  0  -f-  Q 

where  E  and  b  are  the  electromotive  force  and  resistance  of 
the  battery,  respectively.  With  the  switch  K  on  /,  the  galva- 
nometer current  is  given  by  the  equation 


R2 
FIG.  38. 


92       RESISTANCE  AND   ITS  MEASUREMENT     [II,  §  69 


E 


(58) 


Dividing    (57)  by  (58),  and  neglecting  the  resistance  of   the 
battery  as  small,  we  have 


(59) 


d2 


If  g  is  negligible  as  compared  to  the  other  resistances,  we  may 
write 

(60)  ^2  =  ^-^. 

Frequently,  however,  the  available  standard  Rv  is  so  small 
compared  with  Rz  that  the  two  deflections  are  not  readily 
comparable.  For  example,  if  the  value  of  Rl  is  one  tenth 

megohm,  while  that  of  R2  is  100 
megohms,  then  dl  is  one  thousand 
times  as  great  as  d2.  For  such  a 
case  the  deflections  would  not  be  of 
the  same  order  of  accuracy.  In 
order  to  avoid  difficulties  of  this 
kind,  and  to  keep  the  values  of  the 
deflections  nearly  the  same,  the 
galvanometer  is  shunted.  The  fore- 
going method  as  used  with  the 

Ayrton  shunt  is  illustrated  in  Fig.  39,  and  the  formula  will 
now  be  developed.  Suppose  that  for  some  position  a  of  the 
adjustable  contact  arm,  the  value  of  r  is  R/a.  Similarly,  for 
some  position  6,  the  value  of  the  resistance  to  the  left  of  b 
is  R/b.  Calling  Tj  and  72  the  currents  in  the  main  circuit  for 
the  upper  and  lower  positions  of  the  switch,  respectively,  and 
using  equation  (11),  §  26,  we  may  write  expressions  for  the 
corresponding  currents  through  the  galvanometer  in  the  form 

,fin   .  _  pi,  _  r  f  R/a  1      T  T  R/a 

\UJ-y  la  ! 


FIQ.  39. 


II,  §  69]     MEASUREMENT  OF  HIGH  RESISTANCE 

and 

/fi9W       pw  - 

**-  ^- 


Since  Rl  and  7?2  will  always  be  large,  the  currents  through 
them  will  not  be  affected  appreciably  by  a  change  in  the  value 
of  the  shunt  ratio.  Therefore  the  effective  potential  difference 
of  the  battery  is  essentially  unchanged.  We  may  then  write 
equations  (61)  and  (62)  in  the  form 


and 


Dividing  (63)  by  (64),  we  have 


whence 


The  quantities  a  and  b  here  represent  the  fractional  parts  of 
R  corresponding  to  the  positions  of  the  contact  arm.  It  must 
be  kept  in  mind  that  these  fractions  do  not  represent  the  ratio 
of  the  galvanometer  current  to  the  total  line  current,  but  the 
ratio  of  the  two  galvanometer  currents  corresponding  to  the 
positions  a  or  6  and  c.  The  ratio  of  currents  through  the  gal- 
vanometer for  positions  a  or  &  and  c  is  a  constant,  and  is  inde- 
pendent of  the  galvanometer  resistance. 

In  all  these  high  resistance  measurements  we  are  dealing 
essentially  with  materials  other  than  metals.  Nevertheless,  it 
is  assumed  that  Ohm's  law  holds.  This  is  approximately  true 
for  ordinary  insulating  materials  or  for  surface  leakage. 
However,  if  there  is  electrostatic  capacity  present,  as  in  con- 


94        RESISTANCE  AND  ITS  MEASUREMENT     (II,  §  70 

densers  or  long  cables,  special  care  must  be  given  to  the  inter- 
pretation of  the  results. 

Moreover,  in  the  case  of  insulating  substances  it  is  often 
found  that  the  deflection  which  occurs  when  the  electromotive 
force  is  first  impressed  across  the  high  resistance  does  not 
remain  constant,  but  falls  off  at  first  rapidly,  then  more  slowly, 
the  variation  becoming  inappreciable  only  after  a  long  inter- 
val, frequently  several  hours.  This  phenomenon  is  called 
electrification,  and  is  due  to  certain  changes  which  take  place 
within  the  body  of  the  substance.  The  effect  is  less  marked 
as  the  temperature  rises. 

In  order  that  measurements  of  high  resistance  may  have 
definite  significance,  they  must  be  accompanied  by  statements 
of  the  experimental  conditions,  including  impressed  voltage, 
time  of  electrification,  and  temperature.  A  representative 
specification  for  cables  is  as  follows  : 

Each  conductor  shall  show,  after  twenty-four  hours'  immersion  in  water, 
an  insulation  resistance  of  not  less  than  300  megohms  per  mile,  with  not 
less  than  100  volts  applied  for  one  minute,  at  a  temperature  of  75° 
Fahrenheit. 

For  further  details  on  cable  testing  see  §  165,  Chapter  VIII. 

70.  Laboratory  Exercise  XII.  To  measure  a  high  resistance 
by  the  substitution  method. 

APPARATUS.     A  standard  resistance  of  one  tenth  megohm 
.or     more,    sensitive    galvanometer, 
Ayrton  shunt,  well-insulated  switch, 
and  an  E.  M;  F.  of  100  volts  or  more. 
PROCEDURE.     (1)  Arrange  the  cir- 
cuit  as    in   Fig.   39,   with   a   high- 
insulation   switch    in    the    battery 
circuit.      In   case   the  E.  M.  F.   is 
taken  from  a  dynamo,  it  is  well  to 
FIG.  39  (repeated).  include  a  carbon  filament  lamp  in 


II,  §  71J     MEASUREMENT  OF  HIGH  RESISTANCE       95 

the  circuit  in  order  to  prevent  an  excess  of  current  in  case  of 
an  accidental  short-circuit.  The  resistance  of  this  lamp  may 
be  known  and  allowed  for,  or  it  may  be  neglected  if  the 
resistance  to  be  measured  is  high  enough.  It  is  always  pref- 
erable to  use  a  well-insulated  battery  for  the  E.  M.  F. 

(2)  The  maximum  current  will  pass  through  the  galvanome- 
ter when  the  switch  is  on  point  1,  Fig.  15,  §  27,  and  the  first 
position  to  try  is  then  on  the  other  side,  beginning  on  the  zero 
point,  and  setting  successively  on  6,  c,  d,  etc.,  until  a  readable 
deflection  occurs. 

In  using  the  Ayrton  shunt  it  must  be  kept  in  mind  that  the  ratios  given 
hold  only  when  the  deflections  for  the  respective  positions  of  the  switch 
are  compared  with  the  deflection  which  would  occur  with  the  switch  on 
point  1.  The  multiplying  factors  for  the  other  positions  are  then  10,  100, 
1000,  etc.  This  differs  from  the  ordinary  shunt  practice,  where  the  mul- 
tiplying factor  gives  the  ratio  between  total  current  and  galvanometer 
current.  In  the  Ayrton  shunt  the  galvanometer  is  always  in  parallel 
with  the  combined  resistances  of  all  the  steps. 

(3)  Read  and  record  the  deflections  for  the  standard  resist- 
ance and  for  the  samples  furnished. 

(4)  Calculate  the  values  of   the  unknowns  by  substituting 
the  proper  values  in  equation  (65).     If  insulation  resistance  of 
wire  or  cable  is  being  found,  express  the  result  in  megohms 
per  mile. 

All  parts  of  the  circuit  must  be  well  insulated,  and  since  the  E.  M.  F. 
used  will,  in  general,  be  high,  the  utmost  precaution  must  be  exercised  to 
avoid  a  dangerous  short-circuit.  In  case  porcelain  or  glass  insulators  are 
being  tested,  the  resistance  of  one  of  them  may  be  too  high  to  be  measured 
conveniently.  A  group  of  ten  or  a  hundred  are  then  connected  in  paral- 
lel, the  resistance  of  a  single  one  being  inferred  from  the  result. 

71.  The  Loss  of  Charge  or  Leakage  Method.  For  some 
kinds  of  testing  where  the  resistance  of  cables,  dielectric  resist- 
ance of  condensers,  etc.,  is  to  be  measured,  the  loss  of  charge 
method  is  preferred.  This  method  involves  the  use  of  the 
ballistic  galvanometer,  and  it  will  be  described  after  this 
instrument  has  been  studied.  See  Chapter  VIII,  §  165. 


96       RESISTANCE  AND   ITS  MEASUREMENT     [II,  §  72 


PART  V.     THE  MEASUREMENT  OF  LIQUID  RESISTANCE 

72.  Resistance  of  Electrolytes.  With  respect  to  electrical 
resistance,  liquids  may  be  divided  into  two  classes : 

I.  Good  conductors. 

II.  Poor  conductors,  commonly  called  non-conductors. 
Under  the  first  class  two  groups  are  recognized  :  (a)  mercury 

and  other  fused  metals  ;  (6)  electrolytes,  which  are,  in  general, 
water  solutions  of  salts  and  acids. 

Of  the  liquids  classed  as  good  conductors,  those  in  group  (a) 
offer  no  difficulty  so  far  as  the  measurement  of  their  resistance 
is  concerned.  The  Wheatstone  bridge  or  any  fall  of  potential 
method  yields  accurate  results. 

Under  group  (6),  however,  special  difficulties  are  encountered. 
An  electrolyte  conducts  by  virtue  of  the  decomposition  of  the 


FIG.  40. 

solution  and  the  migration  of  the  ions,  and  there  is,  in  gen- 
eral, a  double  procession  of  ions  in  opposite  directions,  the 
ions  finally  being  deposited  on  the  electrodes.  This  sets  up 
a  counter  electromotive  force  of  polarization  which  affects 
measurements  as  though  an  extra  resistance  had  been  added. 
Hence,  determinations  based  on  Ohm's  law  are  inaccurate. 

The  Wheatstone  bridge  may  be  used,  however,  by  putting 
the  sample  of  the  electrolyte  to  be  measured  in  a  suitable  tube 
or  vessel  provided  with  platinum  electrodes  (Fig.  40),  'connect- 
ing this  as  the  unknown  arm  of  the  bridge,  and  impressing  an 


II,  §  72]    MEASUREMENT  OF   LIQUID   RESISTANCE     97 

alternating  instead  of  a  steady  E.  M.  F.  A  sensitive  telephone 
receiver  will  replace  the  galvanometer,  and  will  give  a  mini- 
mum sound  when  the  bridge  is  balanced.  If  the  alternating 
current  is  supplied  from  the  secondary  of  a  small  induction 
coil  from  which  the  condenser  has  been  removed,  the  polariza- 
tion during  one  half  of  the  cycle  will  be  annulled  by  the 
E.  M.  F.  of  opposite  sign  during  the  following  half  cycle,  and  a 
balance  may  be  obtained  from  which  the  true  resistance  of  the 
sample  is  found. 

The  electrodes  should  be  coated  with  a  deposit  of  platinum 
black,  which  greatly  increases  the  effective  area,  thus  reducing 
the  surface  density  of  such  residual  charges  as  are  present. 
The  size  and  style  of  the  containing  tubes  will  depend  upon 
the  conductivity  of  the  solution  to  be  measured.  A  poorly 
conducting  electrolyte  should  be  measured,  in  general,  in  a 
short  tube  with  a  large  electrode  area. 

The  tube  selected  is  first  calibrated  by  measuring  in  it  the 
resistance  of  a  sample  of  a  standard  solution  of  known  con- 
ductivity, such  as  a  normal  solution  of  potassium  chloride. 
After  carefully  cleansing  the  tube,  the  sample  of  the  electrolyte 
of  unknown  conductivity  is  measured  in  the  same  way. 

The  resistivity  of  any  substance  is  given  by  the  equation 
(3)  of  §  39.  If  the  length  of  the  column  of  electrolyte  is 
denoted  by  L,  the  resistivities  in  the  two  cases  are  given  by 
the  equations 

(66)  *,  =  .Ri, 


where  ~ki  is  the  resistivity  of  the  standard  solution,  &2  is  that 
of  the  unknown,  R±  and  Rz  are  the  measured  resistances  respec- 
tively, and  a/L  is  the  ratio  of  the  area  of  cross-section  to  the 
length  of  the  column  of  electrolyte. 

Since  this  ratio  is  a  constant  for  the  given  containing  cell, 

H 


98       RESISTANCE  AND   ITS  MEASUREMENT     [II,  §  72 

the  unknown  resistivity  is  found  by  dividing  (66)  by  (67)  and 
solving  for  k2 ;  this  gives 

(68)  *i-*if- 

Ml 

The  conductivity  is  the  reciprocal  of  resistivity  and  is  ex- 
pressed in  terms  of  the  reciprocal  ohm  or  the  mho  (§  40).  A 
table  of  conductivities  for  potassium  chloride  at  different  tem- 
peratures is  given  in  the  appendix. 

If  polarization  effects  are  avoided,  Ohm's  law  holds  through- 
out a  wide  range  of  values  for  electrolytes,  from  highly  con- 
ducting salt  solution  to  poorly  conducting  water.  The  con- 
ductivity of  pure  water  has  been  measured  in  a  vacuum  and 
found  to  be  0.4  x  10~6  mhos  per  centimeter  cube.  Water  in 
contact  with  the  atmosphere  has  a  conductivity  about  twice  as 
great,  and  this  cannot  be  reduced  because  of  the  tendency  to 
dissolve  carbonic  acid  and  ammonia  from  the  air,  as  well  as 
certain  substances  from  the  glass  vessel  in  which  it  is  usually 
kept.  While  the  amount  of  these  impurities  is  very  minute, 
the  effect  on  the  conductivity  of  the  water  is  comparatively 
very  large. 

Precise  measurements  require  that  the  applied  alternating 
wave  of  E.  M.  F.  shall  have  the  form  of  a  sine  curve,  with 
equal  positive  and  negative  amplitudes,  and  that  the  bridge 
arms  shall  be  free  from  capacity  and  inductance. 

Except  for  the  highest  accuracy,  the  above  method,  due  to 
Kohlrausch,  is  satisfactory.  Such  measurements  are  of  par- 
ticular interest  in  physical  chemistry,  in  affording  informa- 
tion as  to  the  degree  to  which  any  dissolved  substance  has 
been  dissociated. 

73.  Laboratory  Exercise  XIII.  To  find  the  conductivity  of 
an  electrolyte. 

APPARATUS.  Wheatstone  bridge,  sensitive  telephone  re- 
ceiver, small  induction  coil,  resistance  box,  one  or  two  dry 
cells,  containing  cell,  and  thermometer. 


II,  §  73]    MEASUREMENT  OF  LIQUID   RESISTANCE     99 

A  convenient  form  of  Wheatstone  bridge,  known  as  the  Kohlrausch 
bridge,  is  shown  in  Fig.  41.  A  uniform  manganin  wire  nearly  five  meters 
long  is  wound  in  ten  turns  on  a  marble  cylinder.  This  is  covered  with  a 
protecting  hood  which  revolves  on  a  vertical  spindle,  threaded  with  a 


FIG.  41. 

pitch  exactly  equal  to  that  of  the  winding  of  the  wire.  A  spring  brush 
fastened  to  the  inside  of  the  hood  is  always  in  contact  with  the  wire, 
and  is  connected  through  the  spindle  to  the  binding  post  on  the  front  of 
the  base.  The  two  ends  of  the  wire  are  attached  to  the  two  binding 
posts  at  the  back.  Total  turns  are  read  on  the  glass  scale,  and  fractions 
of  a  turn  are  read  on  the  graduated  circle  against  a  vertical  line  etched 
on  the  glass  plate.  Ratios  of  lengths  on  either  side  of  the  contact  point, 
and  hence  also  ratios  of  the  resistances  of  wire  segments,  are  readily 
determined. 


PROCEDURE.  (1)  Draw  the  usual  diamond  diagram  of  the 
Wheatstone  bridge,  properly  label  the  arms,  and  write  the 
corresponding  formula. 

The  entire  arrangement  of  the  circuit,  with  the  Kohlrausch 
bridge  inserted,  is  shown  in  Fig.  42,  p.  100. 


100     RESISTANCE  AND   ITS   MEASUREMENT     [II,  §  73 

(2)  Connect  the  containing  cell  E  in  series  with  the  known 
resistance  and  with  the  bridge  wire,  as  in  Fig.  42.  Attach  the 
telephone  and  the  secondary  of  the  induction  coil  to  the  proper 
terminals.  Usually  it  is  preferable  to  connect  the  telephone 
receiver  between  A  and  B,  although  either  of  the  conjugate 


FIG.  42. 

arrangements  may  be  used.  Compare  this  circuit  with  the 
diagram  previously  drawn,  and  write  the  formula  required  for 
this  actual  arrangement. 

(3)  Measure  a  sample  of  the  standard  solution,  adjusting  R 
and  the  position  of  the  contact  on  the  bridge  wire  until  the 
minimum  sound  is  found  near  the  middle  of  the  wire.     Take 
the  temperature  of  the  sample. 

(4)  Rinse  out  the  cell  and  refill  it  with  the  electrolyte  of 
unknown  conductivity.     Measure  its  resistance  as  above  and 
take  its  temperature. 

(5)  Calculate  the  resistivity  from  equation  (68)  and  find  the 
conductivity  of  the  samples  at  the  observed  temperature. 

(6)  If  the  balance  point  is  not  sharply  denned,  the  electrode 
surfaces  probably  need  replatinizing.     These  surfaces  may  be 
rinsed  with  clear  water  after  using,  but  they  should  not  be 
touched  with  the  fingers  nor  wiped  with  a  towel. 

An  increase  in  sensitiveness  may  be  effected  by  putting  a  short-circuit- 
ing key  across  the  telephone  terminals  and  repeatedly  making  and  break- 
ing the  circuit  for  any  particular  setting  of  the  movable  contact.  The 
ear  is  more  sensitive  to  sudden  changes  than  to  gradual  ones. 


CHAPTER   III 

ELECTROMOTIVE   FORCE   AND   POTENTIAL 
DIFFERENCE 

PART  I.     SOURCES  OF  ELECTROMOTIVE  FORCE 

74.  There   are   three  well-known  classes  of  apparatus  for 
transforming  energy  of  other  kinds  into  electrical  energy.     Of 
these,  the  thermoelectric  generator  is  of  use  only  for  relatively 
feeble  currents.     Its  phenomena  and  some  of  its  applications 
will  be  discussed  in  Art.  81.     The  dynamo  generator  depends 
upon  principles  which  will  be  discussed  in  Chapter  VI. 

In  the  following  articles  some  of  the  more  common  types 
of  battery  cells  will  be  described,  as  these  are  especially  im- 
portant in  connection  with  electrical  measurements.  The 
function  of  a  battery  cell  is  to  transform  chemical  energy  into 
electrical  energy.  Two  classes  of  cells  will  be  considered, 
primary  batteries,  which  are  renewed  by  the  addition  of  fresh 
chemicals,  and  secondary  batteries,  or  accumulators,  which  are 
renewed  or  charged  by  means  of  a  current  which  is  applied  in  a 
direction  opposite  to  that  yielded  by  the  cell  when  in  service, 
and  which  produces  in  the  cell  the  necessary  chemical  changes. 

75.  The  Gravity  Cell.     The  gravity  cell  has  for  its  negative 
electrode  a  zinc  plate,  suspended  near  the  top  of  the  contain- 
ing jar,  and  for  its  positive  electrode  a  plate  of  copper  at  the 
bottom  of  the  jar.    This  plate  is  covered  with  a  layer  of.  copper 
sulphate  crystals,  and  over  it  for  a  few  inches  is  a  water  solu- 
tion of  the  same  salt.     Above  this,  and  surrounding  the  zinc 
plate,  is  a  solution  of  zinc  sulphate.     A  sharp  line  of  separa- 

101 


102  ELECTROMOTIVE  FORCE  [III,  §  75 

tion  should  appear  between  the  two  solutions.  Due  to  its 
greater  density  the  copper  sulphate  solution  tends  at  first  to 
remain  below  the  zinc  sulphate.  However,  diffusion  will  occur 
unless  the  battery  is  supplying  current.  This  type  yields  the 
most  constant  electromotive  force  of  any  of  the  ordinary 
primary  cells.  Owing  to  its  freedom  from  polarization,  it 
is  useful  for  closed  circuit  work.  Its  E.  M.  F.  is  1.08  volts, 
and  its  internal  resistance  may  vary  from  half  an  ohm  to 
several  ohms. 

To  set  up  a  gravity  cell  for  experimental  purposes,  first  fill 
the  jar  half  full  of  water  and  stir  in  about  50  grams  of  ZnS04. 
Place  the  copper  in  position  at  the  bottom  of  the  cell  and  hang 
the  zinc  at  the  top.  By  means  of  a  funnel  tube  introduce  a 
saturated  solution  of  CuS04  at  the  bottom  of  the  cell,  taking 
every  precaution  against  mixing  the  two  solutions.  The  blue 
should  appear  at  the  bottom,  and  gradually  rise,  carrying  the 
white  ZnS04  solution  above  it.  Continue  introducing  CuS04 
solution  until  the  zinc  plate  is  well  covered  by  the  ZriS04.  In 
case  the  fluids  mix  they  must  be  emptied  out  and  a  fresh  trial 
made.  When  through  using  a  gravity  cell,  if  it  is  desired  to 
keep  it  set  up,  close  its  circuit  through  a  resistance  of  a  few 
hundred  ohms,  in  order  to  prevent  diffusion  of  the  solution. 
If  it  is  not  to  be  kept  set  up,  pour  out  the  solution  into  the 
waste  jar,  rinse  the  parts  in  clean  water,  and  wipe  them  dry 
with  a  towel.  These  battery  solutions  should  not  be  emptied 
into  the  ordinary  laboratory  sinks. 

76.  The  Leclanche  Cell.  The  Leclanche  cell  has  a  voltage 
of  1.4,  a  low  internal  resistance,  and  rapid  polarization.  It  is 
useful  chiefly  for  open-circuit  work.  The  elements  are  zinc 
and  carbon  in  a  solution  of  salammoniac.  The  chemical  action 
is  complex,  a  double  chloride  of  zinc  and  ammonium  being 
formed,  with  liberation  of  hydrogen  and  ammonia  gas  at  the 
carbon  plate.  To  diminish  polarization  effects,  a  paste  con- 


Ill,  §  77]     SOURCES  OF  ELECTROMOTIVE  FORCE     103 

taining  black  oxide  of  manganese  is  used,  sometimes  packed 
in  a  porous  jar  about  the  carbon  electrode,  and  sometimes 
formed  into  solid  bars,  which  are  held  close  to  the  carbon  plate 
by  rubber  bands.  Recovery  from  polarization  is  rapid  when 
thus  treated. 

77.  The  Dry  Cell.  The  so-called  dry  cell  is  a  modification 
of  the  Leclanche  cell,  in  which  the  solutions  are  absorbed  in 
the  materials  of  the  cell,  thus  rendering  it  portable  in  any 
position,  with  no  liquids  to  be  spilled.  The  E.  M.  F.  is  1.5-1.6 
volts,  and  when  fresh,  its  internal  resistance  is  low,  so  that 
with  an  external  resistance  of  0.01  ohm,  a  current  of  25  or  30 
amperes  can  be  drawn  from  it. 

The  anode  is  a  sheet  of  zinc  in  cylindrical  form,  which  is 
the  outer  casing  or  container  for  the  cell.  This  zinc  shell  is 
lined  with  several  layers  of  porous  paper  heavily  impregnated 
with  ammonium  chloride  solution.  Within  this  paper  is  a 
granular  mixture  of  carbon,  manganese  dioxide,  and  ammonium 
chloride  saturated  with  zinc  chloride  and  ammonium  chloride 
solutions.  Packed  securely  within  this  mixture  and  at  the 
center  of  the  cell  is  a  fluted  carbon  rod,  and  the  whole  is  sealed 
air  tight  with  pitch  or  hard  wax. 

Polarization  is  rapid  when  large  currents  are  drawn  from  it, 
but  the  manganese  dioxide  is  a  strong  oxidizing  agent,  and  the 
return  to  nearly  normal  voltage  is  rapid  after  use. 

The  internal  resistance  of  this  type  of  cell  increases  rapidly 
with  use  and  age,  partly  due  to  the  drying  out  of  the  contained 
moisture  and  partly  due  to  the  reduction  of  the  manganese 
dioxide.  There  is  also  a  deposit  on  the  zinc  of  non-soluble 
impurities  and  products  of  secondary  reactions.  Particularly 
in  the  dry  cell,  the  so-called  internal  resistance  is  a  compound 
quantity,  which  is  dependent  upon  many  factors,  and  which 
varies  with  the  current  output,  the  age,  and  the  temperature. 
Ordinary  tests,  such  as  those  described  in  §  §  85-86,  are  of  little 


104  ELECTROMOTIVE  FORCE  [III,  §  7fr 

significance  with,  dry  cells.     They  should  be  subjected  to  serv- 
ice tests,  based  on  the  actual  conditions  of  use.1 

Where  galvanometer  deflections  are  to  be  observed,  constant 
potential  cells  are  necessary.  For  zero  methods,  however,  other 
types  will  often  answer.  Dry  cells  in  series  with  a  high  resistance 
may  be  regarded  as  constant  potential  cells  for  many  purposes. 

78.  The  Edison-Lalande  Cell.    Modifications  of  the  origi- 
nal Lalande  cell,  usually  sold  under  the  name  Edison-Lalande, 
have   an  E.  M.  F.-  of  0.8-0.9   volt,  and  a  low  internal  resist- 
ance of  only  a  few  hundredths  of  an  ohm.     The  negative  elec- 
trode is  amalgamated  zinc,  and  the  positive  electrode  consists 
of  a  plate  of  compressed  copper  oxide,  the  surface  of  which 
has  been  reduced  to  the  metallic  state.     These  plates  are  im- 
mersed in  a  twenty  per  cent  solution  of  sodium  hydroxide.     A 
thin  layer  of  heavy  mineral  oil  is  poured  over  the  top  to  pre- 
vent evaporation  and  creeping.     After  the  cell  has  been  in  opera- 
tion for  a  short  time  the  E.  M.  F.  becomes  practically  constant. 

79.  Standard  Cells.     It  is  convenient  to  have  at  hand  cer- 
tain voltaic   or   electrochemical   standards  of  E.  M.  F.     Two 
such  standards,  the  Clark  cell  and  the  Weston  cell,  can  be  pre- 
pared with  a  high  degree  of  constancy  and  trustworthiness. 

The  Clark  cell  has  zinc  or  zinc  amalgam  for  its  negative, 
and  mercury  for  its  positive  electrode,  the  zinc  being  sur- 
rounded by  a  saturated  solution  of  ZnS04,  while  a  paste  of 
mercurous  sulphate  is  above  the  mercury.  The  E.  M.  F.  of  a 
cell  of  this  sort  is  1.434  volts  at  15°  C.  Between  10°  and  25° 
its  variation  is  about  0.00115  volt  per  degree,  the  E.  M.  F. 
decreasing  with  a  rise  in  temperature. 

The  E.  M.  F.  of  these  cells  will  vary  with  the  concentration 
of  the  solutions  and  with  the  circumstances  of  manufacture. 
Hence  a  certificate  should  accompany  each  cell  when  purchased. 

1  For  standard  methods  of  testing  dry  cells  see  TRANS.  AM.  ELECTRO- 
CHEMICAL SOCIETY,  vol.  21,  1912,  p.  275. 


Ill,  §  79]     SOURCES  OF  ELECTROMOTIVE  FORCE     105 


Wax 


=--=•     E£?£  ZnSO, 


=^=-rr^^£?E  Paste 


This  cell,  as  originally  prepared,  was  not  sufficiently  por- 
table. Moreover,  its  E.  M.  F.  did  not  follow  accurately  the 
changes  in  temperature.  An  improved  form  was  devised  by 
Carhart.  In  this  form  the  zinc  sulphate  solution  is  saturated 
at  0°  C.,  and  its  temperature  coefficient  is  somewhat  less  than 
in  the  earlier  forms.  The  value  of  its  E.  M.  F. 
for  any  temperature  t°  is  found  from  the  formula 

(1)       Et  =  1.440  -  0.00056(£  - 15)  volts. 

The  interior  arrangement  of  a  Carhart-Clark 
cell l  is  shown  in  Fig.  43. 

The  Weston  cell,  or  cadmium  cell,  takes  its 
name  from  the  use  of  cadmium  instead  of  zinc 
as  in  the  Clark  cell.  It  was  first  suggested  by 
Weston  in  1891.  It  is  preferably  made  in  the 
H  form,  as  shown  in  Fig.  44.  Platinum  wires 
are  sealed  into  the  tubes  and  make  contact  with 
mercury  on  the  positive  side  and  with  a  cad- 
mium-mercury amalgam  on  the  negative  side.  Above  the 
mercury  is  a  layer  of  a  thick  paste,  made  by  intimately  mix- 
ing metallic  mercury,  mercurous  sulphate,  and  a  saturated 

solution  of  cadmium  sulphate. 
Wax  Above  the  cadmium  amalgam  on 
the  negative  side  is  a  layer  of 
cadmium  sulphate  crystals,  and 
over  all,  filling  both  sides  of  the 
tube,  is  a  saturated  or  nearly  sat- 
urated solution  of  cadmium  sul- 
phate.  The  open  ends  of  the 
tube  are  closed  with  corks  and 
sealed  with  wax.  When  these 
cells  are  made  according  to  precise  specifications,2  they  can  be 

1  Specifications  for  setting  up  these  cells  are  given  in  the  U.  S.  BUREAU 
OF  STANDARDS  BULLETIN,  vol.  4,  p.  1,  1907. 

2  U.  S.  BUREAU  OF  STANDARDS  BULLETIN,  vol.  4,  p.  1,  1907. 


Fia.  44. 


106  ELECTROMOTIVE  FORCE  [III,  §  79 

reproduced  with  a  variation  of  only  a  few  parts  in  100,000. 
Temperature  changes  have  but  slight  influence  on  this  type  of 
cell,  a  change  of  ten  degrees  C.  either  way  causing  a  change 
in  E.  M.  F.  of  less  than  five  parts  in  10,000.*  For  this  reason 
the  cadmium  cell  has  practically  displaced  all  other  types.  It 
is  portable  and  may  be  sent  through  the  mails  or  otherwise 
transported  without  ill  effects. 

Formerly  the  international  volt  was  defined  as  a  stated  frac- 
tion of  the  E.  M.  F.  of  a  cell  of  the  Clark  type,  but  its  evalua- 
tion depended  upon  the  use  of  a  known  resistance  and  a 
known  current.  The  London  Conference  (1908),  although  de- 
fining the  international  volt  in  terms  of  the  ampere  and  the 
ohm,  recommended  the  adoption  of  the  Weston  cell  as  a  sub- 
standard of  voltage ;  its  value  is  now  taken  as 

1.0183  volts  at  20°  C. 

Standard  cells  are  not  intended  to  supply  current,  and  they 
must  always  be  used  in  such  a  way  that  there  is  no  danger  of 
polarization  taking  place.  They  should  be  used  only  in  series 
with  a  high  resistance  (10,000  ohms),  or  for  zero  methods 
where  the  potential  difference  is  compensated. 

EXERCISE 

Will  a  standard  cell  show  its  rated  voltage  when  connected  to  a  com- 
mercial voltmeter  ?  Explain.  What  would  be  the  effect  on  the  cell  of 
this  experiment  ? 

80.  Secondary  Batteries.  When  a  direct  current  is  passed 
through  an  electrolytic  cell,  decomposition  products  are  de- 
posited on  the  cathode.  In  general,  if  the  source  of  current 
is  removed  and  the  circuit  is  again  completed,  there  will  be  a 
flow  of  current  in  the  reverse  direction,  due  to  the  polariza- 

1  The  precise  formula  from  which  the  E.  M.  F.  of  a  Weston  cell  is  calcu- 
lated for  any  temperature  t°  is 

Et  =  .£'20  —  0.0000406  (t  —  20)  —  0.00000095(<  -  20) 2  +  0.00000001  (t  -  20)8 
See  U.  S.  BUREAU  OF  STANDARDS,  Circular  No.  29. 


Ill,  §  80]     SOURCES  OF  ELECTROMOTIVE  FORCE     107 

tion  of  the  electrodes.-  In  this  way  an  exhausted  battery  can 
have  its  active  material  renewed  by  electrolytic  deposition. 
In  a  class  of  batteries  called  storage  cells,  or  accumulators,  this 
is  an  efficient  process. 

A  common  type  of  accumulator  consists  of  lead  plates  per- 
forated with  many  apertures,  into  which  the  active  material 
is  compressed.  This  is  usually  a  paste  made  by  mixing  cer- 
tain lead  salts  (red  lead,  Pb304,  and  litharge,  PbO)  with  sul- 
phuric acid.  If  plates  thus  prepared  are  immersed  in  a 
twenty  per  cent  solution  of  sulphuric  acid,  and  a  current  is 
sent  through  the  cell,  hydrogen  passes  to  the  cathode  and 
reduces  the  paste  to  spongy  metallic  lead.  The  S04  ions  pass 
to  the  anode,  and  a  higher  oxide  of  lead,  Pb02,  is  formed. 
The  rapid  evolution  of  hydrogen  at  the  cathode  is  evidence  of 
complete  transformation  of  the  material,  and  the  cell  is  then 
said  to  be  charged.  If  the  charging  current  is  now  cut  off  and 
the  cell  connected  to  a  circuit,  current  will  be  found  to  flow 
from  the  cell  in  a  direction  opposite  to  that  during  the  process 
of  being  charged.  Such  cells  have  a  discharge  voltage  of 
about  2.2,  and  a  low  internal  resistance  of  the  order  of  a  few 
hundredths  of  an  ohm.  Moreover,  their  behavior  when  in 
good  condition  is  very  constant,  and  they  are  indispensable 
for  many  kinds  of  electric  testing.  The  disadvantages  of 
this  type  of  cell  are  its  great  weight  and  its  requirement  of 
regular  and  systematic  attention  during  charging  and  use. 

In  the  Edison  storage  cell,  the  positive  plate  consists  of 
nickel  oxide  packed  in  perforated  steel  tubes,  several  of  which 
are  mounted  side  by  side  in  a  steel  frame.  The  negative  plate 
consists  of  iron  oxide  held  in  a  somewhat  similar  way.  These 
plates  are  immersed  in  a  twenty  per  cent  solution  of  caustic 
potash,  and  are  sealed  into  a  container  of  welded  sheet  steel. 
These  cells  have  an  average  voltage  of  1.2,  which  is  a  disad- 
vantage as  compared  with  the  lead  type  of  cell.  However, 
they  are  rugged,  constant  in  their  behavior,  and  but  slightly 


108  .  ELECTROMOTIVE  FORCE  [III,  §  80 

affected  by  extreme  temperatures.  Because  of  their  smaller 
weight,  they  are  preferred  for  many  purposes  to  other 
types. 

The  storage  cell  does  not  store  electricity,  but  by  means  of 
the  current  supplied  to  it  chemical  changes  are  set  up  which 
renew  the  active  material  necessary  for  the  continued  operation 
of  the  cell. 

81.  Thermoelectricity.  In  1826  Seebeck  discovered  that  a 
current  of  electricity  flows  in  a  circuit  consisting  of  two  differ- 
ent metals,  when  a  difference  of  temperature  is  maintained  at 
the  two  points  of  contact,  or  junctions.  This  is  commonly 
explained  by  saying  that  at  the  junction  thermal  energy  is 
transformed  into  electrical  energy,  and  this  point  is  regarded 
as  the  seat  of  an  E.  M.  F. 

In  the  following  table  some  common  metals  are  so  arranged 
that  when  any  two  are  chosen  for  the  circuit,  current  flows 
across  the  hot  junction  from  any  metal  to  one  standing  lower 

in  the  list. 

Bismuth  Lead 

Nickel  Silver 

Platinum  Iron 

Copper  Antimony 

Of  the  pure  metals  the  bismuth-antimony  pair  yields  the 
greatest  thermoelectromotive  force,  that  is,  these  elements 
have  the  greatest  thermoelectric  power.  However,  certain 
alloys  such  as  german  silver,  advance,  and  platinum  with  irid- 
ium  or  rhodium,  are  frequently  used  for  one  of  the  materials. 
Expressed  in  microvolts  per  degree  of  temperature  difference, 
the  thermoelectromotive  force  of  a  pair  of  iron-german  silver 
junctions  is  about  25,  that  of  an  iron-advance  pair  is  about  as 
great,  and  that  of  a  copper-advance  pair  is  about  40.  The 
purity  and  the  physical  state  of  these  materials  is  an  impor- 
tant factor  in  the  measured  values. 


Copper 


Copper 


ill,  §  81]     SOURCES  OF  ELECTROMOTIVE  FORCE     109 

The  thermoelement  is  made  by  soldering  two  pieces  of  the 
chosen  metals  together,  with  their  ends  soldered  respectively 
to  pieces  of  copper  wire.  The  junctions  AA,  Fig.  45,  are  kept 
at  some  constant  temperature,  usually  that  of  melting  ice,  and 
the  junction  B  is  Iron  Advance 

heated.  The  gal- 
vanometer  will 
then  indicate  the 
passage  of  a  cur- 
rent. Knowing  FlG-  45' 
the  sensibility  of  the  galvanometer  and  the  resistance  of  the 
circuit,  the  effectiveness  of  the  arrangement  in  microvolts  per 
degree  can  be  calculated.  The  presence  at  the  junction  of  an 
intermediate  metal  or  alloy  like  solder  will  not  affect  the  value 
of  the  E.  M.'F.,  because  whatever  effect  is  developed  at  one 
point  of  contact  is  annulled  at  the  other. 

Many  and  varied  forms  of  thermoelements  find  application 
in  the  measurement  of  temperatures  where  mercury-in-glass 
thermometers  would  be  too  massive  to  respond  quickly  to 
small  temperature  changes,  or  where  it  would  be  impossible 
or  inconvenient  to  introduce  them.  The  chief  advantage  of 
thermoelements  is  found  in  their  small  mass  and  their  quick 
response  to  changes  of  temperature.  The  range  over  which 
they  may  be  used  is  limited  only  by  the  temperature  at  which 
the  metals  oxidize,  or  for  some  metals,  the  temperature  at 
which  the  effect  changes  sign.  For  example,  in  the  iron-copper 
element  the  effect  changes  sign  at  about  270°  C.  For  a  range 
from  liquid  air  temperatures,  -  190°  C.  to  200°  or  300°  C., 
copper-advance  or  iron-german  silver  is  used.  For  higher 
temperatures,  upwards  of  1700°  C.,  a  thermocouple  of  platinum 
and  a  platinum-rhodium  alloy  is  used.  The  E.  M.  F.  may  be 
measured  by  means  of  a  potential  galvanometer l  or  by  a  po- 

1  Since  the  resistance  of  the  thermocouple  and  its  circuit  is  usually  low, 
the  galvanometer  should  also  be  of  low  resistance. 


110 


ELECTROMOTIVE  FORCE 


[HI,  §  81 


Fia.  46. 


tentiometer  method.     Any  thermoelement  may  be  calibrated 
with  a  galvanometer  so  that  temperatures  can  be  read  directly. 

82.   Laboratory  Exercise  XIV.     To  calibrate  a  galvanometer 
used  with  a  copper-advance  thermoelement. 

APPARATUS.     Galvanometer,   thermocouples  with   hot  and 
cold  baths  and  thermometers,  one  accumulator  cell,  voltmeter, 

two  resistance  boxes,  and  a  tap 
key. 

PROCEDURE.  (1)  Arrange  the 
circuit  as  shown  in  Fig.  46,  keep- 
ing the  junction  B  at  the  tem- 
perature of  melting  ice.  Heat 
the  junction  A  by  means  of  a 
flame  under  the  oil  bath,  and  take  simultaneous  readings  of 
temperatures  and  galvanometer  deflections.  A  multiplier  of 
required  amount  to  keep  the  deflection  on  the  scale  for  the 
highest  temperature  used  (about  200°  C.)  should  be  put  in 
series  with  the  galvanometer.  Take  eight  or  ten  readings  over 
approximately  equal  intervals  of 
temperature  change,  from  room 
temperature  to  200°  C. 

(2)  Plot  deflections  as  ordinates 
and  temperature  readings  as  ab- 
scissas on  a  sheet  of  cross-section 
paper. 

(3)  To  calibrate  the  galvanom- 
eter in  microvolts  per  scale  di- 
vision, connect  the  apparatus  as  in  Fig.  47.     A  storage  cell  8 
is  put  in  series  with  a  resistance  of  1000  ohms.     Traveling 
contacts  at  aa'  will  enable  a  potential  difference  one  thousandth 
of  that  of  S  to  be  applied  to  the  galvanometer  in  series  with 
the  variable  resistance  R.     The  voltage  of  the  cell  may  be 
taken  with  a  voltmeter.     Take  deflections,  five  or  more  in 


FIG.  47. 


Ill,  §  83]     SOURCES  OF  ELECTROMOTIVE  FORCE     111 


number,   over  the  full-scale   range.     Calculate  the  potential 
difference  at  the  galvanometer  terminals  in  microvolts. 

(4)  Use  the  same  sheet  as  in  (2)  above,  and  lay  off  a  scale 
of  microvolts  along  the  axis  of  abscissas.  Plot  deflections 
against  microvolts.  From  the  two  curves  thus  plotted,  calcu- 
late the  potential  differences  developed  at  the  various  temper- 
atures, and  plot  a  third  curve  showing  the  relation  between 
potential  difference  and  temperature. 

83.  Electromotive  Force  and  Terminal  Potential  Dif- 
ference. In  general,  whenever  two  plates  of  different  metals 
are  dipped  in  an  electrolyte,  a  difference  of  potential  is  found 
to  exist  between  them.  If  dilute  sulphuric  acid  is  used  as  the 
liquid,  the  potential  relations  of  a  few  familiar  substances  are 
given  in  the  following  table.  Selecting  any  two  of  the  sub- 
stances, current  is  observed  to  flow  through  a  wire  connecting 
them  outside  of  the  battery,  from  any  one,  to  one  which  stands 
lower  in  the  table. 


As  substances,  these  are 
electropositive  upward. 


Carbon 

Mercury 

Copper 

Iron 

Cadmium 

Zinc 


As  battery  poles,  these 
are  positive  down- 
ward. 


Zn 


The  terms  positive  and  negative  as  applied  to  the  poles 
of  a  battery  must  not  be  confused  with  electropositive  and 
electronegative  as  applied  to  the  substances. 
The  seat  of  the  E.  M.  F.  is  at  the  surface  of 
contact  between  the  electropositive  substance 
(zinc)  and  the  liquid,  and  the  negative  pole  4 
is  the  exposed  part  of  the  electropositive  plate.  \ 
In  a  simple  voltaic  cell  (Fig.  48)  charges  of 
opposite  sign  are  found  at  the  poles.  If  the 
poles  are  connected  by  a  conductor,  these  charges  are  removed. 


FIG.  48. 


112 


ELECTROMOTIVE  FORCE 


[III,  §  83 


However,  the  action  of  the  cell  restores  them  and  the  potential 
difference  is  maintained.  There  is,  then,  some  process  going 
on  within  the  cell  which  tends  to  maintain  this  transfer  of 
charge  from  the  negative  pole  to  the  positive  pole  within  the 
cell,  and  from  positive  to  negative  outside.  This  is  called  the 
electromotive  force,  or  E.  M.  F.  of  the  cell. 

In  the  external  conductor,  as  well  as  in  the  liquid  of  the 
cell,  there  will  be  a  fall  of  potential  in  the  direction  of  the 
current  flow,  and  at  the  interface  between  the  zinc  plate  and 
the  liquid  there  will  be  a  rise  in  potential,  due  to  a  transfor- 

/  V 


Zn 


FIG.  49. 


mation  of  chemical  energy.  The  potential-resistance  relations 
throughout  the  entire  circuit  are  illustrated  in  Fig.  49.1  At 
the  zinc-liquid  surface  there  is  a  sudden  rise  in  potential  ah. 
Through  the  liquid  resistance  b  there  is  a  fall  of  potential  fe. 
Also,  there  is  a  fall  of  potential  ed  at  the  liquid-copper  surface, 
and  a  further  drop  dc  through  the  external  resistance  R.  In 
this  figure,  vertical  distances  represent  potential  differences, 
and  horizontal  distances  represent  resistances  ;  hence,  it  is 
clear  that  the  slope  of  the  lines  he  and  da1  is  measured  by  the 
ratio  of  potential  difference  to  resistance  respectively.  This 
slope  is  then  equal  to  the  current  flowing,  and  since  the  cur- 

1  Figure  49  represents  a  special  case  where  the  line  ga',  parallel  to  he,  inter- 
cepts the  line  ah  at  the  point  g,  so  that  gh  =fe.  In  general  this  line  will  not 
cut  ah  at  g,  but  at  some  point  which  may  be  called  g'.  The  student  should 
draw  other  similar  figures  with  R  much  larger  and  much  smaller  than  the 
battery  resistance. 


Ill,  §83]     SOURCES  OF  ELECTROMOTIVE  FORCE     113 


rent  is  constant  in  all  parts  of  the  circuit,  the  angles  6  and  0' 
are  equal.  The  maximum  effective  potential  difference  J5J,  or 
the  total  E.  M.  F.,  is  given  by  the  equation 

ah 


(2) 


=  cd-\-  ef, 


while  the  potential  difference  effective  in  the  external  part  of 
the  circuit,  or  E',  is  represented  by  cd. 
From  similar  triangles  it  then  follows  that 


or,  by  composition, 

whence 
(3) 
or 
(4) 


ef  _ac_  m 
cd     ca'  ' 

cd  +  ef  _  ac  +  ca'  _  aar  . 


cd 


ca' 


ca 


7  ' 


^-  = 

E' 
E'  = 


E 


R 


The  quantity  E'  is  called  the  terminal  potential  difference  and 
it  is  seen  that  E'  will  approach  E  as  b  diminishes  in  value,  or 
as  R  becomes  so  great  that  in  comparison  with  it  b  may  be 
disregarded. 

When  a  potential  galvanometer  is  connected  across  the 
terminals  of  a  cell  which  is  not  delivering  current,  the  ob- 
served reading  E  is  the  E.  M.  F.  of  the  cell,  or 
the  open-circuit  potential  difference.  If,  with- 
out removing  the  galvanometer,  a  resistance 
R,  Fig.  50,  is  connected  across  the  battery 
terminals,  a  current  will  flow,  and  the  gal- 
vanometer reading  E'  will  be  less  than  before. 
This  decrease  will  go  on  as  the  value  of  R  is 
made  less,  until  when  R  =  0,  that  is,  when  the 
cell  is  short-circuited,  the  galvanometer  will  show  no  deflection 
whatever.  The  potential  galvanometer  shows,  for  any  value 


114  ELECTROMOTIVE  FORCE  [III,  §  83 

of  72,  the  then  existing  potential  difference  at  the  cell  terminals,  and 
this  may  vary  from  the  value  of  the  E.  M.  F.  of  the  cell  to  zero. 
For  any  value  of  R  the  current  flowing  is  given  by  Ohm's  law 
in  the  form 

®  ':=CT 

or 

(6)  E  =  ib  +  iR. 

The  term  iR  is  equal  to  E\  whence 

(7)  E'  =  E  -  ib. 

Solving  equation  (7)  for  6,  the  internal  resistance  of  the  cell, 
we  find 

(8)  6  =  ^^; 

1 

77" 

and  since  i  =  — ,  equation  (8)  may  be  written  in  the  form 
R 


Since  galvanometer  deflections  are  proportional  to  potential 
differences,  equation  (9)  may  be  written  in  the  form 

(10)  a 


The  preceding  equations  hold  not  only  for  the  voltaic  cell,  but 
also  for  any  other  form  of  generator.  The  battery  resistance 
will  not  be  constant  for  all  values  of  current,  and  in  stating 
the  value  of  the  internal  resistance,  the  current  output  must 
be  specified. 

The  E.  M.  F.  may  be  expressed  in  terms  of  the  work  done 
in  conveying  a  unit  charge  once  around  the  entire  circuit. 
Referring  to  Fig.  49,  the  work  done  by  the  cell  at  the  inter- 
face between  the  zinc  plate  and  the  liquid  is  proportional  to 
line  ah.  On  the  same  scale,  fe  represents  the  work  done  by 
the  current  against  the  ohmic  resistance  of  the  electrolyte. 


Ill,  §  84]     SOURCES  OF  ELECTROMOTIVE  FORCE     115 

The  current  does  work  proportional  to  ed  at  the  liquid-copper 
surface,  and  a  further  amount  of  work  proportional  to  dc  is 
done  in  the  external  resistance  R.  For  one  such  cycle  the 
battery  must  supply  an  amount  of  energy  proportional  to  the 
difference  between  ah  and  ed,  or  cd  +  ef.  The  work  available 
for  the  external  circuit  is  only  that  which  is  proportional  to 
cd. 

The  entire  circuit,  including  the  battery  resistance,  may  be 
regarded  as  divided  into  small  portions.  Then  across  the  ter- 
minals of  each  portion  there  will  be  a  certain  potential  differ- 
ence, or  potential  drop,  which  will  vary  for  any  given  portion 
with  the  value  of  the  current  flowing.  The  algebraic  sum  of  all 
these  differences  of  potential  over  the  entire  circuit  will  give 
the  value  of  the  E.  M.  F. 

In  order  to  avoid  confusion,  it  is  customary  to  use  the  term 
potential  difference,  or  potential  drop,  with  reference  to  certain 
limited  portions  of  the  circuit,  and  to  reserve  the  expression 
electromotive  force  for  that  maximum  value  of  the  terminal 
potential  difference  which  the  generator  yields  when  measured 
with  no  current  flowing.  Invariably  when  a  current  is  flowing, 
some  part  of  the  E.  M.  F.  is  required  to  overcome  the  effective 
internal  resistance  of  the  generator,  and  the  available  potential 
difference  of  the  terminals  is  always  less  than  the  E.  M.  F.  by 
this  amount. 

84.  Battery  Resistance.  It  is  important  to  note  that  the 
quantity  b,  which  has  been  called  the  internal  resistance  of 
the  battery,  is  not  a  constant,  but  varies  more  or  less  with  the 
current  drawn  from  the  cell.  Some  batteries  (e.g.  the  gravity 
type)  show  a  decreasing  resistance  with  an  increase  in  current. 
Some  have  a  high  polarization  (e.g.  the  dry  cells),  which  tends 
to  increase  with  the  current  output.  The  back  E.  M.  F.  of 
polarization  opposes  the  E.  M.  F.  of  the  cell,  and  the  resultant 
or  effective  potential  difference  is  decreased.  The  effect  is 


116  ELECTROMOTIVE  FORCE  [III,  §84 

the  same  as  that  of  an  increased  internal  resistance,  and  the 
values  of  b  calculated  from  equation  (8)  are  apparently 
increased. 

The  temperature  coefficient  of  the  internal  resistance  is 
large,  arid  the  total  ampere-hour  output  of  the  cell  also  affects 
its  value.  The  quantity  b  is  an  important  one,  and  it  is  treated 
as  a  resistance  ;  properly  it  should  be  called  the  effective  resist- 
ance of  the  cell.  A  complete  study  of  a  battery  cell  involves 
the  determination  of  several  factors  which  will  be  considered 
again  in  §  89. 

85.  Laboratory  Exercise  XV.  To  study  the  variation  in  the 
potential  difference  at  the  battery  terminals,  and  to  find  the  internal 
resistance  of  the  battery  cell  by  the  galvanometer  method. 

APPARATUS.  Potential  galvanometer  with  shunt  and  high- 
series  resistance,  reversing  switch,  resistance  box,  tap  key,  and 
one  gravity  cell. 

PROCEDURE  (1).  Arrange  the  circuit  as  in  Fig.  51.  Adjust 
the  galvanometer  to  zero  and  choose  such  values  for  the  shunt 
S  and  the  series  resistance  Rf  that  the 
deflection  d^  will  be  about  full  scale 
with  K  open.  The  series  resistance 
should  be  at  least  a  thousand  ohms. 

(2)  Make  R  =  200  ohms,  close  the 
key  K,  and  read  the  deflection  d2. 
Make  R  smaller,  decreasing  by  ten 
"""I  ^  ~~\  steps  until  zero  is  reached,  choosing 
' */vs/ww\* — '  the  steps  so  that  the  deflections  de- 
crease by  approximately  equal  inter- 
vals, and  read  the  deflection  for  each  step.  The  connecting 
wires  from  R  to  the  battery  should  be  as  short  as  possible. 
Why? 

(3)  Tabulate  values  of  R,  right  and  left  galvanometer  deflec- 
tions, and  mean  deflections.  The  deflection  dl  is  proportional 


Ill,  §  86]     SOURCES  OF  ELECTROMOTIVE  FORCE     117 

to  the  E.  M.  F.  of  the  cell.  The  other  deflections  are  proportional 
to  the  respective  values  of  the  terminal  potential  difference. 

(4)  Plot  a  curve  with  values  of  R  as  abscissas,  and  deflec- 
tions as  ordinates,  d±  being  the  maximum  value  of  the  series. 
State  what  inference  may  be  drawn  from  this  curve.     Locate 
on  the  ;y-axis  a  point  at  d,/2.     Project  this  point  on  the  curve 
and  read  the  corresponding  value  of  R.     Show  that  this  is  ap- 
proximately the  battery  resistance. 

(5)  Calculate  from  equation  (10)  the  value  of  the  battery 
resistance,  choosing  several  different  values  of  d2.     If  this  is 
not  a  constant,  account  for  its  variation. 

86.   Laboratory  Exercise  XVI.     To  measure  the  internal  re- 
sistance of  a  battery  by  the  voltmeter-ammeter  method. 

APPARATUS.     Voltmeter,  ammeter,  battery  to  be  tested,  con- 
trol resistance,  and  tap  key. 

PROCEDURE.  (1)  Arrange  the  circuit  as  in  Fig.  52.  R  is  a 
control  rheostat,  the  resistance  of  which  need  not  be  known. 
With  K  open,  read  E  on  the  voltmeter, 
which  gives  the  E.  M.  F.  of  the  cell.  Close 
Kj  having  adjusted  R  so  that  the  range 
of  the  ammeter  is  not  exceeded,  and  again 
take  the  voltmeter  reading  E',  simulta- 
neously reading  the  current  i. 

(2)  Take  several  sets  of  E  and  i  values, 

and  calculate  the  value  of  b  from  equation 

FIG.  52. 
(8).     In  case  the  battery  consists  of  more 

than  one  cell,  for  example,  a  storage  battery,  the  value  of  b 
determined  above  is  the  sum  of  the  internal  resistances  of 
the  several  cells.  The  mean  value  for  one  cell  is,  however, 
readily  calculated. 

(3)  If  storage  batteries  are  under  test,  E  may  be  the  poten- 
tial difference  measured  while  the  battery  is  being  charged, 
the  charging  current  i  being  read  at  the  same  time.     Then, 


118  ELECTROMOTIVE  FORCE  [III,  §  86 

with  the  charging  current  cut  off,  E'  is  the  terminal  potential 
difference.  The  internal  resistance  is  then  given  by  equation  (8) 
The  difference  between  the  two  voltages  is  that  required  to  send 
the  current  i  through  the  internal  resistance  of  the  battery. 

87.   Laboratory  Exercise  XVII.     To  compare  electromotive 
forces  by  the  condenser  method. 

APPARATUS.     Ballistic   galvanometer,    standard    condenser, 
three-way   discharge   key,   standard   cell,  and   battery   to   be 

tested.  The  theory  of  the  ballistic 
galvanometer  and  condenser  will  be 
given  in  subsequent  chapters. 

PROCEDURE.  (1)  Arrange  the  cir- 
cuit as  in  Fig.  53,  with  a  standard 
cell  at  B.  Charge  the  condenser  by 
pressing  the  key  to  6,  then  discharge 

it  by  raising  the  key  to  a.  and  read 
FIG.  53.  J  J 

the  deflection  d^ 

(2)  Replace  the  standard  cell  by  the  cell  to  be  tested  and 
repeat  (1),  reading  the  deflection  dz. 

(3)  When  the  condenser  is  charged,  the  quantity  is  given 
by  the  equations 

(H)  Q1  =  Odl=GVl, 

and 

(12)  Q2=GA=CF2, 

where  Qi  and  Q2  are  the  quantities  stored  in  the  condenser 
when  charging  potentials  Fi  and  Vz  are  impressed.  The  cor- 
responding deflections  are  d^  and  d2  respectively,  and  G  is  the 
constant  of  the  galvanometer.  Then,  from  (11)  and  (12),  we  have 


(4)  Calculate   the   value   of    the    unknown    E.  M.  F.   from 
equation  (13). 


Ill,  §  89]     SOURCES  OF  ELECTROMOTIVE  FORCE     119 

88.   Laboratory  Exercise   XVIII.     To  measure  the  internal 
resistance  of  a  battery  by  the  condenser  method. 

APPARATUS.     As  in  Laboratory  Exercise  XVII,  §  87,  together 
with  a  resistance  box  and  a  tap  key. 

PROCEDURE.     (1)  Arrange  the  circuit  as  in  Fig.  54.     With 
k  open,  press  K  to   b  for  an  instant  and  then  raise  it  to  a. 
Read  the  deflection  on  the  gal- 
vanometer €?. 


I 

r 


(2)  With  k  closed,  set  R  at 
some  low  value,  and  repeat  the 
procedure  of  (1),  reading  the 
deflection  d2.     Repeat  for  sev- 
eral different  values  of  R. 

(3)  Calculate  the  value  of  b 

from  equation  (10).     Each  de-  Fi     ^ 

flection   used   in   the    formula 

should  be  the  mean  of  several  observations.     Tabulate  all  data 

and  results. 

This  method  is  most  reliable  when  the  battery  is  one  which 
does  not  polarize  rapidly,  and  which  has  a  high  internal  re- 
sistance. If  a  standard  condenser  of  low  value  is  used,  the 
charge  flowing  into  it  will  not  appreciably  polarize  the  cell. 

89.  Test  of  a  Primary  Cell.  The  two  chief  characteristics 
of  a  battery  cell  are  its  E.  M.  F.  and  its  internal  resistance. 
In  general,  that  cell  is  most  useful  in  which  the  E.  M.  F.  is 
high  and  the  internal  resistance  low.  There  are  three  differ- 
ent tests  to  which  a  battery  must  be  subjected  in  order  to 
investigate  systematically  and  completely  its  quality  and  use- 
fulness.1 These  are:  (1)  the  time  test,  (2)  the  life  test,  and 
(3)  the  efficiency  test. 

The  time  test  shows:  (a)  the  decrease  in  E.M.F.  due  to 
polarization,  together  with  the  rate  of  this  decrease  when  the 

1  For  standard  methods  of  testing  dry  cells,  see  PBOC.  AM.  ELECTRO- 
CHEMICAL SOCIETY,  vol.  21,  p.  275,  1912. 


120  ELECTROMOTIVE  FORCE  [III,  §  89 

cell  is  kept  for  a  given  period  on  closed  circuit  through  a 
specified  resistance ;  (b)  the  rate  and  extent  of  recovery  from 
polarization ;  (c)  the  terminal  potential  difference  when  the 
cell  is  closed  through  a  fixed  resistance. 

The  life  test  is  quite  the  same  as  the  foregoing,  except  that 
the  polarization  is  allowed  to  continue  until  the  E.  M.  F.  has 
been  reduced  to  at  least  half  of  its  initial  value,  after  which 
the  recovery  is  observed  for  a  similar  period. 

The  efficiency  test  shows :  (a)  the  ratio  of  the  quantity  of 
electricity  obtained  from  the  cell  by  the  consumption  of  a 
given  mass  of  zinc  to  the  quantity  necessary  to  deposit  the 
same  mass  in  an  electrolytic  cell ;  (b)  the  ratio  of  the  energy 
available  in  the  external  circuit  to  that  dissipated  within  the 
cell  itself  as  heat.  If  these  last  two  tests  are  carried  out, 
they  will  work  the  cell  to  exhaustion.  If  the  first  of  them 
is  carried  over  a  period  of  perhaps  an  hour,  it  will  give  an 
accurate  indication  of  the  intrinsic  worth  of  the  cell.  Hence 
it  is  the  more  common,  and  is  frequently  the  only  test  made. 

90.  Laboratory  Exercise  XIX.  To  make  a  time  test  of  a 
battery  cell. 

APPARATUS.     As  in  Laboratory  Exercise  XVIII,  §  88,   to- 
gether with  a  watch. 

PROCEDURE.  (1)  Arrange  the 
circuit  as  in  Fig.  55.  If  a  Le- 
clanche  cell  is  used,  make  R  about 
five  ohms ;  if  a  dry  cell  is  used, 
make  R  about  half  an  ohm.  With 
~ |  k  open,  charge  and  discharge  the 

s/WW  condenser  by  successively  throw- 

ing K  to  b  and  a.     The  observed 
deflection  will  be  proportional  to  the  E.  M.  F.  of  the  cell. 

(2)  The  E.  M.  F.  of  the  cell  may  be  expressed  in  volts  by 
means  of  a  calibration  curve  for  the  galvanometer,  which  is 


Ill,  §  90]     SOURCES  OF  ELECTROMOTIVE  FORCE     121 

prepared  as  follows.  Before  beginning  the  test  on  the  battery 
B,  put  in  its  place  a  standard  cell,  and  note  the  galvanometer 
deflection  d,  using  the  same  value  of  the  capacity  as  that 
which  will  be  used  throughout  the  test.  On  a  sheet  of  cross- 
section  paper,  scale  off  one  axis  in  volts  and  the  other  in  de- 
flections, and  locate  the  point  corresponding  to  d  and  the 
E.  M.  F.  of  the  standard  cell.  This  will  be  one  point  on  the 
calibration  curve.  If  the  deflections  are  strictly  proportional 
to  quantities  of  electricity,  and  hence  to  charging  potentials, 
the  curve  will  be  a  straight  line  passing  through  the  origin. 
From  this  curve  any  value  of  the  deflection  may  be  read  directly 
in  volts.  The  scales  should  be  so  chosen  that  at  least  hun- 
dredths  of  a  volt  can  be  read. 

(3)  Observing  the    exact   time,   close   k    and   immediately 
charge  and  discharge  C  as  before,  observing  the  deflection  c?2. 
The  terminal  potential  difference  can  then  be  taken  from  the 
calibration  curve.     Keeping  k  closed,  again  read  d2  after  an 
interval  of  two  minutes.     After  four   minutes,  open  k  for  a 
very  short  time,  just  long  enough  to  manipulate  the  key  K, 
and  observe  another  value  of  d1}  which  gives  the  open  circuit 
voltage  at  that  time.     After  six  minutes  take  a  reading  of  d2 
as  before,  and  after  eight  minutes  with  k  open  for  an  instant, 
take  another  reading  for  dv 

Continue  in  this  way  for  one  hour,  taking  readings  of  the 
open-circuit  voltage  every  four  minutes,  in  order  to  get  the 
decrease  due  to  polarization.  Alternate  with  these,  also  at  four 
minute  intervals,  readings  for  the  terminal  potential  difference. 
The  key  k  will  be  left  firmly  closed  except  at  the  appropriate 
four-minute  intervals,  when  it  is  opened  for  a  second  or  two  in 
order  to  secure  the  open-circuit  readings.  A  convenient  form 
for  ft  is  a  spring  tap  key,  making  contact  on  the  upper  points. 

(4)  At  the  end  of  the  hour,  open  k  and  continue  the  read- 
ings as  before,  in  order  to  determine  the  rate  and  the  extent 
of  the  recovery  from  polarization.     For  the  first  quarter  of  an 


122  ELECTROMOTIVE  FORCE  [III,  §  90 

hour  of  recovery  readings  should  be  taken  every  two  or  three 
minutes.  For  the  rest  of  the  period,  or  until  the  recovery  curve 
ceases  to  rise,  longer  intervals  will  suffice.  Avoid  closing  the 
circuit  containing  the  battery  under  test  until  quite  ready  to 
begin  counting  time. 

(5)  Calculate  from  equation  (10)  the  internal  resistance  of 
the  cell  for  ten  or  more  sets  of  values  of  E  and  E'.     Also  cal- 
culate the  same  number  of  values  of  the  current  strength  by 
dividing  the  respective  values  of  the  terminal  potential  differ- 
ence by  the  external  resistance  R. 

(6)  Tabulate  in  full  the  values  of  time,  dlt  dz,  open-circuit 
voltage,  terminal  potential  difference,  internal  resistance,  and 
current. 

Tabulate,  also,  values  of  time,  deflections,  and  voltages  dur- 
ing the  recovery  period. 

(7)  On  a  sheet  of  squared  paper  choose  a  suitable  time  scale 
along  the  aj-axis,  and  along  the  y-axis  arrange  three  scales  of 
suitable  range  for  voltage,  resistance,  and  current.     Plot  five 
curves :    (1)  open-circuit   E.  M.  F.,    (2)  recovery,   (3)  terminal 
potential  difference,  (4)    internal  resistance,  (5)    current.     It 
is  customary  to  start  the  recovery  curve  at  the  last  reading  of 
the  polarization  curve,  running  it  toward  the  left,  above  the 
polarization  curve. 

Discuss  the  curves  and  state  what  inferences  may  be  drawn 
as  to  the  excellence  of  the  cell. 


Ill,  §  91]  POTENTIOMETERS 


PART  II.     POTENTIOMETERS 


123 


91.  General  Principles  and  Simple  Circuit.  Perhaps  no 
single  instrument  is  capable  of  more  general  application  in  the 
electrical  laboratory  than  the  potentiometer,  which  is,  as  its 
name  implies,  a  device  for  measuring  potential  differences. 
A  simple  form  of  potentiometer  circuit  is  shown  in  Fig.  56. 

W.S. 


FIG.  56. 

A  uniform,  homogeneous  wire  PP',  usually  a  meter  or  more 
in  length,  is  stretched  over  a  graduated  scale,  and  connected 
in  series  with  a  constant  battery  WB,  called  the  working 
battery.  An  adjustable  resistance  R  is  introduced  for  the 
purpose  of  controlling  the  potential  difference  between  P  and 
P'.  This  resistance,  as  well  as  the  wire,  must  be  so  chosen  as 
to  carry  the  necessary  current  without  sensible  heating.  A 
double-pole,  double-throw  switch  U  enables  either  a  standard 
cell  S  or  the  test  cell  E  to  be  connected  to  the  points  W\. 

Let  us  suppose  (1)  that  the  potential  difference  between  P 
and  P'  is  greater  than  that  of  the  cell  S,  and  (2)  that  the 
circuit  is  so  arranged  that  the  poles  of  the  working  battery 
and  S  are  opposed.  Then,  if  the  switch  U  is  on  the  points  1, 
1,  it  will  be  possible  to  find  two  points  Fand  Vi  on  PP',  such 
that  the  fall  of  potential  between  them  is  just  equal  to  the 
potential  difference  at  the  terminals  of  S.  In  this  case  no 


124 


POTENTIAL  DIFFERENCE 


[III,  §  91 


current  will  flow  through  the  galvanometer  g,  and  the  circuit 
is  said  to  be  compensated  or  balanced. 

If  the  current  flowing  through  the  wire  is  constant,  the 
potential  difference  between  V  and  V\  is  proportional  to  the 
resistance  of  the  wire  included  between  these  points,  and  if 
the  wire  is  uniform  and  homogeneous,  the  resistance  is  propor- 
tional to  the  length.  Hence,  the  distance  d^  between  V  and 
Fj,  read  on  the  scale  under  the  wire,  may  be  assumed  to  be 
proportional  to  the  value  of  the  potential  difference  at  the 
terminals  of  the  standard  cell  S. 

If  the  switch  U  is  thrown  across  the  points  2,  2,  any  other 
potential  difference,  such  as  E,  may  be  balanced  against  the 

W.B. 


FIG.  56  (repeated). 

fall  of  potential  along  some  other  length  of  the  wire  d2.     Then 
we  have  the  relation 

(W)  f-J, 

whence  the  value  of  E  is  found  to  be 


(15) 


. 
dl 


The  method  may  be  simplified  and  the  apparatus  made  to 
read  directly  by  the  following  procedure.  Fix  V  at  the  point 
P,  and  graduate  the  scale  into  1500  parts,  with  the  zero  at  P 


Ill,  §  92]  POTENTIOMETERS  125 

and  the  1500  mark  at  P'.  Assume  that  we  are  using  a  Clark 
cell,  for  which  the  voltage,  corrected  for  temperature,  is  1.432. 
Set  the  point  Vl  at  the  scale  division  1432,  taking  care  that 
the  conditions  1  and  2,  as  given  above,  are  fulfilled.  Adjust 
R  until  the  galvanometer  shows  no  deflection.  Then  the 
potential  drop  along  the  wire,  which  is  directly  proportional  to 
the  distance  from  P  as  read  on  the  scale,  is  just  equal  to  the 
known  voltage  of  the  standard  cell.  Assuming  the  constancy 
of  this  adjustment,  values  of  potential  difference  may  be  read 
directly  from  the  scale. 

The  great  utility  of  the  potentiometer  lies  in  the  fact  that 
it  may  be  used  to  measure  current  strength  and  resistance,  as 
well  as  potential  difference.  The  strength  of  a  current  is 
determined  by  finding  the  potential  drop  between  the  ter- 
minals of  a  standard  resistance  while  the  current  is  flowing 
through  it.  A  resistance  is  measured  by  comparing  the  poten- 
tial difference  across  the  unknown  resistance  with  that  across 
a  known  resistance,  both  carrying  the  same  unvarying  current. 

It  is  thus  seen  that  all  of  these  measurements  are  really 
referred  to  the  E.  M.  F.  of  a  standard  cell  and  a  standard  re- 
sistance. These  two  quantities  have  been  so  thoroughly  studied 
that  the  greatest  confidence  is  felt  in  their  correctness  and 
permanence,  provided  that  temperature  corrections  are  prop- 
erly applied. 

A  distinct  advantage  of  the  method  is  that  the  standard  cell 
is  so  used  that  at  the  moment  of  balance  no  current  whatever 
is  being  drawn  from  it.  Hence  precise  results  uninfluenced  by 
polarization  can  be  obtained.  Furthermore,  at  the  instant  of 
balance,  the  lead  wires  do  not  carry  any  current,  so  that  errors 
due  to  potential  drop  or  contact  resistance  do  not  occur. 

92.  A  Resistance-box  Potentiometer.  The  long  wire  of  the 
simple  potentiometer  may  be  replaced  by  a  pair  of  resistance 
boxes,  in  which  case  a  greater  degree  of  precision  may  be  attained. 


126 


POTENTIAL  DIFFERENCE 


[HI,  §  92 


The  circuit  of  Fig.  57  shows  such  an  arrangement.  The 
working  battery  WE  is  placed  in  series  with  two  resistance 
boxes  R!  and  R2)  each  of  10,000  ohms  or  more.  The  sum  of  R1 
and  R2  must  be  kept  constant.  A  standard  cell  S  with  its 
poles  opposed  to  the  working  battery  is  connected  in  series 


FIG.  57. 

with  a  galvanometer  across  the  terminals  of  R2.  If  the  E.  M.  F. 
of  the  standard  cell  is  less  than  the  potential  drop  along  PP', 
some  value  of  R2  can  be  found  for  which  the  potential  drop 
measured  by  iRz  is  just  equal  to  the  E.  M.  F.  of  S.  For  this 
value  of  R*  there  will  be  no  current  through  the  galvanometer. 
For  every  change  made  in  R2  an  equal  compensating  change 
must  be  made  in  R±. 

It  is  convenient  to  use  two  boxes  with  the  same  range  for  Rl 
and  R2,  and  also  to  start  with  all  the  plugs  out  of  Rl  and  with 
R2  =  0,  increasing  Rz  until  a  balance  is  found.  A  high  re- 
sistance of  10,000  ohms  should  be  placed  in  series  with  S, 
and  the  key  K  should  be  tapped  cautiously,  until  it  is  seen 
that  the  deflection  is  not  going  to  exceed  the  range  of  the  scale. 
When  a  balance  is  nearly  reached,  the  high  resistance  may  be 
removed. 

With  suitably  chosen  resistance  boxes,  a  high  degree  of  pre- 
cision is  possible.  The  double  adjustment  is  tedious,  however, 
and  more  convenient  arrangements  are  found  in  the  commer- 
cial types,  which  are  described  in  §  §  94  and  96. 

93.  Laboratory  Exercise  XX.  To  measure  electromotive 
force  with  the  simple  potentiometer  and  with  the  resistance-box 
potentiometer. 


Ill,  §  93]  POTENTIOMETERS  127 

APPARATUS.  Storage  battery  and  control  rheostat,  long 
wire  on  baseboard  with  scale,  standard  cell,  cell  to  be  tested, 
three  resistance  boxes,  galvanometer,  connecting  wire,  and  tap 
key. 

PROCEDURE.  (1)  Arrange  the  circuit  as  in  Fig.  56,  with  the 
standard  cell  in  series  with  the  galvanometer.  Tap  the  con- 
tact key  at  extreme  ends  of  the  wire  and  note  whether  the 
galvanometer  reverses.  If  no  reversal  occurs,  interchange  the 
terminals  of  the  standard  cell,  or  increase  the  voltage  at  PP'. 
Find  the  point  on  the  wire  for  which  no  deflection  occurs,  and 
read  this  position  on  the  scale.  Replace  the  standard  cell  by 
the  cell  to  be  measured  and  again  read  the  position  of  the 
contact  for  no  deflection.  The  ratio  of  these  two  readings 
will  give  the  ratio  of  the  electromotive  forces  of  the  two  cells. 

(2)  Make  the  potentiometer  direct  reading  as  explained  in 
§  91,  and  again  measure  the  E.  M.  F.  of  the  test  cell. 

(3)  Note  carefully  by  how  much  the  position  of  the  contact 
point  can  be  shifted  without  disturbing  the  balance,  and  state 
the  probable  precision  of  the  settings.      Each  determination 
should  be  the  mean  of  several  readings. 

(4)  In  place  of  the  long  wire,  connect  two  similar  resistance 
boxes  (Fig.  57),  and  remove  all  the  plugs  from  one  box,  say  RI. 
With  the  galvanometer  and  standard  cell  connected  across  R2,  tap 
the  key  and  note  the  deflection.     Make  R±  low  and  R2  high,  again 
tap  the  key,  and  note  whether  the  galvanometer  reverses  its  de- 
flection.    If  a  reversal  does  not  occur,  the  circuit  is  not  properly 
arranged.    Adjust  R1  and  R2  until  the  deflection  is  zero  when  the 
key  is  tapped,  and  read  R2.    Replace  the  standard  cell  by  the  test 
cell  and  repeat  the  procedure.     The  ratio  of  the  values  of  Rz 
will  give  the  ratio  of  the  electromotive  forces  of  the  two  cells. 

(5)  'As  in  (3)  above,  note  how  much  R2  must  be  changed  in 
order  to  cause  the  least  observable  deflection  on  the  galvanom- 
eter.    After   calculating   the  unknown  E.  M.  F.,  discuss   the 
precision  of  the  results  by  the  various  methods. 


128 


POTENTIAL  DIFFERENCE 


[III,  §  94 


94.  The  Wolff  Potentiometer.  The  circuit  of  the  Wolff 
potentiometer  is  shown  in  Fig.  58,  and  a  conventionalized  dia- 
gram of  it  is  shown  in  Fig.  59.  In  order  to  understand  this 
circuit,  refer  again  to  Fig.  56,  and  fix  clearly  in  mind  the  fol- 
lowing points : 


Standard  Cell 
© 


P     O 


(1)  The  potential  drop  between  P  and  P'  must  be  greater 
than  the  potential  difference  to  be  measured. 

(2)  The  terminals  of  the  same  sign  must  be  connected  to 
the  same  side  of  the  circuit. 

(3)  However  the  contact  points  VV\  are  moved,  or  wher- 
ever they  are  applied,  along  PP',  the  total  resistance  of  PP', 
and   also   the   current    strength   through  PP',   must   remain 
constant. 

In  Fig.  59,  WB  is  the  working  battery,  R  is  a  control 
resistance,  and  PP'  can  be  readily  traced.  The  upper  and 
lower  contact  points,  represented  by  the  arrowheads,  are 
connected  mechanically,  but  they  are  electrically  insulated 
from  each  other.  When  any  one  of  the  switches  F2,  F3,  or  F4 
is  moved  to  the  right,  the  effective  resistance  between  Fand 
FI  is  diminished.  The  total  resistance  PP'  is  kept  constant, 
however,  since  a  compensating  increase  is  effected  by  the  slid- 


Ill,  §  94]  POTENTIOMETERS  129 

ing  of  the  upper  arrowheads  along  the  resistance  wire.  On 
the  other  hand,  if  any  one  of  the  switches  F2,  F3,  or  F4  is 
moved  to  the  left,  the  effective  resistance  between  F  and  FI 
is  increased ;  but  the  total  resistance  PPr  is  kept  constant  by 
a  compensating  decrease  along  the  upper  branch.  With  this 


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V2 


FIG.  59. 


arrangement,  we  can  keep  PP'  constant,  while  varying  the 
potential  difference  between  F  and  FI  through  any  value  from 
zero  to  the  maximum  potential  drop  along  PP'. 

The  circuit  of  this  potentiometer,  as  actually  arranged  for 
use,  is  shown  in  Fig.  58.  The  corresponding  points  in  Figs. 
58  and  59  are  similarly  lettered.  The  working  battery  and 
galvanometer  will  be  connected  as  indicated,  with  the  standard 
cell  and  the  unknown  potential  difference  at  E  and  at  X, 
respectively.  Either  E  or  X  may  be  thrown  into  the  circuit 
by  the  switch,  as  desired.  In  series  with  the  working  battery 
is  a  control  resistance  adjustable  to  one  tenth  ohm  or  less. 
Any  sensitive  galvanometer  will  suffice,  although  for  rapid 
work  a  critically  damped  d'Arsonval  is  most  convenient.  The 
working  battery  should  be  one  or  more  storage  cells  in  good 
condition,  and  carefully  insulated.  Dry  cells  may  also  be 
used,  inasmuch  as  the  resistance  in  series  with  them  is  high. 


130  POTENTIAL  DIFFERENCE  [III,  §  95 

95.  Laboratory  Exercise  XXI.  To  compare  electromotive 
forces  with  the  Wolff  potentiometer. 

APPARATUS.  Potentiometer,  storage  battery  or  other  cells, 
control  resistance,  galvanometer,  standard  cell,  and  cell  to  be 
tested. 

It  will  be  seen  by  reference  to  Fig.  58  that  the  resistance 
of  the  potentiometer  coils  in  series  with  the  working  battery 
is  made  up  of  fourteen  1000-ohm  coils,  nine  100-ohm  coils,  nine 
10-ohm  coils,  nine  1-ohm  coils,  and  nine  0.1-ohm  coils,  making 
a  total  of  14,999.9  ohms. 

PROCEDURE.  (1)  Connect  a  two-volt  storage  battery  and  a 
control  resistance  to  the  working  battery  terminals  PP',  with 
a  standard  cell  and  the  test  cell  at  E  and  X,  respectively. 
Insert  the  galvanometer  at  the  place  indicated.  Put  the  battery 
switch  on  E  and  the  galvanometer  switch  on  the  100,000-ohm 
point. 

Note  the  temperature  of  the  standard  cell  and  compute  the 
correction,  if  any,  setting  the  dial  switches  V,  Vi}  Vzt  F3,  and 
F4  to  read  the  corrected  value.  Adjust  the  control  resistance 
until  the  galvanometer  deflection  is  small  when  the  key  is 
tapped. 

Reduce  the  high  resistance  to  10,000  ohms,  or,  if  necessary, 
to  zero,  in  which  case  the  switch  rests  on  the  left-hand  point. 
Then  continue  the  adjustment  of  the  control  resistance  until 
there  is  no  deflection  when  the  key  is  pressed. 

When  this  adjustment  has  been  made,  the  dials  read  the  stand- 
ard cell  voltage  directly,  because  the  working  current  through 
PP'  has  been  adjusted  to  such  a  value  that  the  potential  drop 
along  one  of  the  1000-ohm  coils  is  0.1  volt,  along  one  of  the 
100-ohm  coils  is  0.01  volt,  etc.,  each  of  the  other  dials  reading 
the  next  figure  in  turn. 

(2)  Set  the  battery  switch  at  X  and  adjust  the  dial  switches 
until  the  galvanometer  shows  no  deflection,  using  the  high 
resistance  as  before  until  near  a  balance.  The  value  of  the 


Ill,  § 


POTENTIOMETERS 


131 


potential  difference  at  X  may  then  be  read  directly  to  the  fifth 
place  of  decimals. 

It  is  well  to  check  frequently  the  correctness  of  the  standard 
cell  adjustment. 

With  this  arrangement  the  highest  voltage  that  can  be  measured  is 
1.49999.  However,  if  the  working  battery  voltage  is  made  ten  times  as 
great,  it  is  evident  that  the  potential  drop  in  Frnay  be  made  one  volt  for 
each  1000-ohm  coil,  and  similarly,  the  drop  along  any  other  set  of  coils 
will  be  increased  tenfold.  In  this  case  any  voltage  up  to  14.9999  may 
be  read  directly.  In  case  higher  voltages  than  this  are  to  be  measured, 
a  volt  box  must  be  used,  as  described  in  §  99. 

96.  The  Leeds  and  Northrup  Type  K  Potentiometer.  This 
instrument  is  somewhat  simpler  in  design  than  the  one  de- 


FIG.  GO. 

scribed  in  §  95,  and  has  some  superior  features.  An  exterior 
view  of  the  apparatus  is  shown  in  Fig.  60,  and  the  complete 
working  circuit  is  shown  in  detail  in  Fig.  61.  The  working 
battery  is  shown  at  W,  Fig.  61.  In  series  with  this  are  the 
control  rheostat  R,  a  set  of  resistance  coils  at  D  whose  func- 


132  POTENTIAL  DIFFERENCE  [III,  §  96 

tion  will  be  explained  later,  fifteen  coils  of  exactly  five  ohms 
each,  and  a  manganin  wire  CB,  which  also  has  a  resistance  of 
exactly  five  ohms.  This  wire  is  wound  upon  a  marble  cylinder, 
and  a  sliding  brush  M'  makes  contact  at  any  point  along  it. 
If  now  the  resistance  of  R  is  so  adjusted  that  the  current  from 
W through  DACE  is  1/50  ampere,  there  will  be  a  potential  drop 
of  1/10  volt  across  the  terminals  of  CB,  and  also  across  the 
terminals  of  each  coil  in  AC.  The  wire  CB  is  about  190  inches 
long,  and  it  is  wound  in  ten  turns  about  the  cylinder.  Whence 
there  is  a  potential  drop  in  the  length  of  one  turn  of  0.01  volt. 
By  means  of  the  scale  and  index  on  the  glass  plate  in  front  of 
the  cylinder,  1/100  of  a  turn  is  easily  read,  the  potential  drop 
over  this  portion  of  the  wire  being  0.0001  volt.  One  tenth  of 
this  subdivision,  which  corresponds  to  0.00001  volt,  may  be 
estimated.  The  vertical  scale  on  the  glass  plate  gives  the 
number  of  turns,  and  the  scale  on  the  horizontal  circle  of  the 
revolving  hood  gives  the  fractions  of  a  turn. 

Between  the  points  M  and  M '  (Fig.  61)  the  potential  drop 
is  1.2  volts  plus  the  drop  along  the  wire  from  C  to  M ',  which 
may  be  read,  as  shown  above,  to  0.00001  volt.  The  contact 
points  M  and  M'  are  both  adjustable,  and  may  be  set  on 
extreme  positions  A  and  B,  respectively,  between  which  points 
the  potential  difference  is  1.6  volts.  Now  let  the  points  M 
and  M'  be  ^connected  through  the  right-hand  points  of  the 
double-pole,  double-throw  switch  U,  in  opposing  series  with 
the  E.  M.  F.  to  be  measured  arid  with  a  galvanometer  (Fig.  61). 
Any  potential  difference  up  to  1.6  volts  may  be  compensated, 
the  galvanometer  showing  no  deflection  when  the  drop  along 
the  potentiometer  circuit  equals  the  applied  E.  M.  F.  If  the 
current  has  been  maintained  constant  at  1/50  ampere,  the 
value  of  the  unknown  E.  M.  F.  may  be  read  directly  from 
the  position  of  the  switch  points  M  and  M'. 

To  insure  that  the  current  remains  constant  at  the  required 
value,  a  standard  cell  is  used.  In  Fig.  61,  a  permanent  con- 


Ill,  §  96] 


POTENTIOMETERS 


133 


nection  is  seen  at  0.5,  leading  to  the  switch  U,  thence  to  the 
standard  cell,  and  to  the  point  T.  Between  points  0.5  and  1.5 
there  is  a  potential  difference  of  just  one  volt,  and  the  coils 
to  the  left  of  A  are  arranged  so  that  corrections  may  be  made 
for  slight  changes  in  the  E.  M.  F.  of  the  standard  cell  due  to 
variations  in  temperature. ' 


BA. 


o 


AAMAAAMV 


/A/'               \ 

^  V             /^ 

FIG.  61. 


Referring  again  to  Fig.  61,  suppose  the  plug  to  be  removed 
from  the  socket  marked  1  and  to  be  inserted  in  the  socket  0.1. 
The  effect  of  this  is  to  throw  the  shunt  S  across  the  entire 
potentiometer  circuit,  and  at  the  same  time  to  introduce  at  K 
a  series  resistance.  The  values  of  S  and  K  are  so  chosen  that 
the  total  current  from  W  remains  unchanged.  The  shunt  S 
is  of  such  value  that  the  current  through  DACB  is  just  1/10 
as  great  as  before,  which  means  that  the  potential  drop  across 
any  given  resistance  is  likewise  1/10  as  great  as  before.  This 


134  POTENTIAL  DIFFERENCE  [III,  §  96 

enables  low  potential  differences  to  be  read,  from  0.15  volt 
downward  by  steps  of  0.00001  volt.  Series  resistance  coils 
are  introduced  at  V  to  safeguard  the  galvanometer  against 
an  excess  of  current. 

97.  Laboratory  Exercise  XXII.     To  compare  electromotive 
forces  with  the  type  K  potentiometer. 

APPARATUS.  Potentiometer,  control  rheostat,  and  working 
battery  of  one  or  two  cells,  Weston  standard  cell,  galvanom- 
eter, and  E.  M.  F.  to  be  measured. 

PROCEDURE.  (1)  With  the  plug  switch  in  socket  1  (Fig. 
61),  connect  the  working  battery  and  the  control  rheostat  to 
the  proper  terminals.  Connect  also  the  standard  cell,  the 
E.  M.  F.  to  be  measured,  and  the  galvanometer,  as  shown  in 
Fig.  61. 

(2)  From  the   certificate  accompanying   the   standard   cell 
ascertain  its  correct  voltage  and  set  the  dial  switch  T  to  cor- 
respond.    With  U  and  V  on  the  left-hand  points,  tap  the  key 
and  adjust  R  until  no  deflection  occurs  on  the  galvanometer. 
Make  the  final  adjustment  of  R  with  F"on  the  right-hand  point. 

(3)  Throw  U  to  the  right-hand  points  and  V  to  the  extreme 
left,  tap  the  key,  and  adjust  the  dial  switches  M  and  M*  until 
the  galvanometer  deflection  is  small.     Throw  V  to  the  right 
and  complete  the  adjustment  for  no  deflection.     The  perma- 
nency of  the  correct  working  conditions  should  be  frequently 
checked  by  throwing  U  to  the  left  and  tapping  the  key. 

(4)  For  measuring  a  small  voltage,  less  than  0.15,  change 
the  plug  from  position  1  to  0.1,  and  proceed  as  before. 

(5)  For  high  voltages,  greater  than  1.5,  use  must  be  made 
of  the  volt  box,  as  explained  in  §  99. 

98.  Laboratory  Exercise  XXIII.     To  measure  current  strength 
with  the  potentiometer  and  to  calibrate  an  ammeter. 

APPARATUS.     Potentiometer,  two  separate  storage  batteries, 


Ill,  §  98] 


POTENTIOMETERS 


135 


FIG.  62. 


control  resistance,  galvanometer,  standard  cell,  one  or  more 
standard  resistance  coils,  rheostat,  and  ammeter  to  be  cali- 
brated. 

PROCEDURE.  (1)  The  arrangement  of  the  apparatus  is  as 
shown  in  Fig.  62.  The  battery  B  is  sending  current  through 
an  adjustable  rheostat  M,  an  ammeter 
Am,  and  a  standard  resistance  r, 
which  must  have  a  sufficient  current- 
carrying  capacity  so  that  overheating 
will  not  occur.  The  terminals  of  r 
will  be  connected  to  the  test  circuit 
of  the  potentiometer.  For  some  value 
of  the  current  as  read  on  the  ammeter,  the  potential  difference 
at  the  terminals  of  r  will  be  measured  in  terms  of  a  standard 
cell.  This  value  of  the  potential  difference  divided  by  the 
known  value  of  r  will  give  the  value  of  the  current  flowing. 
If  this  is  not  in  agreement  with  the  ammeter  reading,  the  error 
of  the  instrument  is  apparent. 

(2)  Investigate  in  this  way  the  scale  of  the  ammeter  at  four 
or  five  points. 

Plot  a  correction  curve,  that  is,  a  curve  showing  the  relation 
between  the  observed  ammeter  readings  and  the  instrumental 
corrections. 

In  general,  the  current  through  a  ten-ohm  standard  coil 
should  not  exceed  1/10  ampere ;  through  a  1-ohm  coil,  1 
ampere ;  and  through  a  1/10-ohm  coil,  5  amperes.  These 
values  are  reasonable  if  we  assume  that  the  coil  is  open  to  the 
air.  When  oil  baths  are  used,  the  current  capacity  is  much 
higher. 

Ammeters  of  the  moving-coil   type  are   subject  to  various 

,  errors,  chiefly  due  to  transportation  or  accident,  temperature 

effects,  or  local  magnetic  fields.     Hence,  they  require  frequent 

calibration.     For  this  purpose,  the  potentiometer   method   is 

well  suited. 


136 


POTENTIAL  DIFFERENCE 


[III,  §  99 


99.  The  Volt  Box.  When  a  constant  current  is  flowing 
through  a  resistance,  the  potential  drop  between  any  two 

points  is  directly  propor- 
tional to  the  resistance  in- 
cluded between  these  points. 
.,  If  a  potential  difference  is 
applied  at  AA  (Fig.  63), 

~ 'lo  the  fraction  of  it  which  ex- 

v/ J 

'M      |  ists  across  AB  is  one  tenth 

as  great,  provided  that  the 

resistance  between  A  and  B  is  one  tenth  of  J2.  Accordingly, 
any  desired  fraction  of  the  impressed  voltage  may  be  secured 
by  adjusting  the  contact  point  B.  This  exact  ratio  only  holds 
when  no  current  is  drawn  from  the  derived  circuit. 

Any  good  resistance  box  provided  with  sockets  and  travel- 
ing plugs  can  be  used  as  a  volt  box,  but  it  is  frequently  con- 
venient to  have  special  designs  for  special  purposes.  Two 
such  special  volt  boxes  are  shown  in  the  accompanying  illus- 
trations. In  Fig.  64,  the  resistances  of  a,  b,  and  c  are  respec- 
tively 200,  1800,  and  18,000  ohms,  the  total  resistance  being 
20,000  ohms.  With  the  switch 

1,   the   voltage    at   P    is  \_) 


FIG.  64. 


on 

one   tenth  of  that  impressed 

at   V,  while  with  the  switch 

on  2,  the  voltage  at  P  is  one 

hundredth   of  that   at  V, 

This   is   the   arrangement  of 

the  volt  box  used  with  the  type  K  potentiometer. 

A  slightly  different  arrangement  is  shown  in  Fig.  65.  The 
resistances  of  a,  6,  c,  d,  and  e  are  respectively  0,  100,  900, 
9000,  and  90,000  ohms.  Hence,  with  K  connected  at  p,  the 
derived  voltage  at  P  is  one  tenth  of  that  impressed  at  V.  The 
contact  K  may  be  moved  to  q  or  r,  in  which  case  the  voltage  at 
P  is  .01  or  .001  of  that  impressed  at  V.  This  is  the  cir- 


Ill,  §  100]  POTENTIOMETERS  137 

cuit  for  the  volt  box  used  with  the  Wolff  potentiometer.  It 

differs  from  the  other  type  in   that  the   switch  controls  the 

position  of  the  impressed  voltage  terminals  instead   of  the 
derived  voltage  terminals. 


FIG.  65. 

QUESTION.  Suppose  a  voltmeter  is  placed  across  P  (Fig. 
65),  in  order  to  measure  the  derived  voltage.  Will  the  volt- 
box  ratios  yield  strictly  accurate  results  ?  Are  the  ratios 
strictly  accurate  when  used  with  a  compensation  scheme,  as 
in  the  potentiometer? 

100.  Laboratory  Exercise  XXIV.  To  measure  a  high  volt- 
age with  the  potentiometer,  and  to  calibrate  a  voltmeter. 

APPARATUS.  Potentiometer,  constant  working  battery  and 
control  resistance,  galvanometer,  standard  cell,  volt  box,  volt- 
meter, and  suitable  source  of  E.  M.  F. 

PROCEDURE.  (1)  The  voltage  to  be  measured  is  impressed 
across  the  terminals  V,  Fig.  64  or  Fig.  65,  and  the  terminals  P 
are  connected  to  the  potentiometer  test  circuit.  The  poten- 
tiometer reading  multiplied  by  the  factor  of  the  volt  box  will 
give  the  desired  voltage. 

(2)  Take  readings  for  three  or  more  points  on  the  voltmeter 
scale,  and  plot  a  curve  showing  the  relation  between  true 
volts  and  scale  readings.  This  is  called  a  calibration  curve. 
When  great  precision  is  desired,  it  is  better  to  plot  the  errors 
of  the  scale  as  ordinates  against  observed  volts. 


138  POTENTIAL  DIFFERENCE  [III,  §  101 

101.  The  Comparison  of  Resistances  with  the  Potentiom- 
eter. The  resistance  to  be  measured  is  connected  in  series 

with  a  standard  resist- 
ance, an  unvarying  cur- 
rent is  passed  through 
both  of  them,  and  po- 
tential wires  are  taken 
from  the  terminals  of 
FlG-  6f>-  each  in  succession  to 

the  test  circuit  of  the  potentiometer.  For  a  constant  working 
battery,  the  potentiometer  readings  will  be  respectively  pro- 
portional to  the  resistances,  whether  or  not  the  instrument 
has  been  adjusted  to  read  volts  directly.  For  precise  compari- 
sons the  standard  resistance  and  the  resistance  to  be  measured 
must  be  immersed  in  oil  baths,  and  the  temperature  must  be 
carefully  controlled  and  read. 

For  measuring  high  resistances  or  those  of  medium  value, 
this  method  offers  no  advantage  over  the  Wheatstone  bridge. 
For  small  resistances,  however,  the  advantage  is  great,  since 
it  is  a  zero  method  and  contact  resistances  are  avoided. 

The  accuracy  of  a  standard  resistance  may  be  checked  by 
using  a  circuit  arranged  as  in  Fig.  66.  A  constant  battery  B 
sends  current  through  the  resistance  to  be  tested  r,  and  through 
a  silver  voltameter  V  (§  113)  in  series  with  it.  Potential 
wires  are  taken  from  the  terminals  of  r  to  the  test  circuit  of 
the  potentiometer,  and  the  strength  of  the  current  is  found 
from  the  mass  of  silver  deposited  on  the  cathode.  The  resist- 
ance of  r  is  then  calculated  from  Ohm's  law. 


CHAPTER   IV 
ELECTRIC    CURRENTS 

102.  Current  Strength.  The  phenomenon  of  current  is  the 
phenomenon  of  the  flow  of  electric  charge.  Current  strength 
is  defined  as  the  time  rate  of  flow  of  charge  along  a  conductor. 
If  the  current  is  constant,  the  charge  wlrch  passes  in  time  t 
seconds  is  given  by 

(1)  Q  =  it, 
whence 

(2)  ,-f 

However,  when  it  is  desired  to  examine  in  a  general  way  all 
possible  phases  of  a  changing  state  of  flow,  it  is  necessary  to 
introduce  instantaneous  values.  If  dQ  and  dt  represent  small 
increments  of  charge  and  time,  respectively,  the  instantaneous 
value  of  the  current  strength  is  given  by  the  formula 1 

(3)  ,_£. 

In  any  event,  it  is  by  means  of  the  current  that  energy  is 
transferred  from  the  generator  through  the  circuit,  and 
liberated  in  one  form  or  another,  depending  on  the  devices  and 
equipment  used.  Hence,  the  measurement  of  current  is  a 
fundamental  one  in  electric  work.  Since  it  is  not  easy  to 
measure  directly  the  simultaneous  values  of  charge  and  time 

1  It  is  here  understood  that  the  current  is  the  same  at  the  same  time 
everywhere  throughout  the  circuit;  but  in  a  large  class  of  problems  dealing 
with  variable  currents,  this  is  not  the  case.  For  example,  in  circuits  contain- 
ing capacity,  such  as  transmission  lines,  account  must  be  taken  of  the  rate  of 
variation  of  the  current  strength  with  distance  along  the  conductor. 

139 


140  ELECTRIC   CURRENTS  [IV,  §  103 

in  order  to  find  their  ratio,  other  methods  are  sought,  in  which 
current  strength  is  quantitatively  associated  with  other  phe- 
nomena which  can  be  measured  more  readily.  There  are  four 
such  phenomena  which  always  accompany  the  flow  of  current 
through  a  circuit,  upon  each  of  which  methods  of  measure- 
ment are  based : 

(a)  The  fall  of  potential  through  a  constant  resistance  of 
known  value,  included  in  the  circuit. 

(b)  The  force  action  of  the  magnetic  field  surrounding  the 
current,  on  other  magnetic  fields. 

(c)  The  heating  effect,  which   appears  when   the   terminal 
device  transforms  the  energy  of  the  current  into  heat. 

(d)  The  electrolytic  effect,  which  occurs  when  the  current 
causes  a  deposit  of  ions  on  the  cathode  of  an  electrolytic  cell. 

103.  Fall  of  Potential.     Whenever  a  current  flows  through 
a  conductor  of  constant  and  known  resistance,  there  is  a  definite 
value  of  the  potential  difference  at  its  terminals,  which  may  be 
measured   with  a   voltmeter.     The   value   of   the   current  is 
readily  found  by  Ohm's  law,  and  is 

(4)  *=-. 

H 

This  is  one  of  the  simplest  methods  of  measuring  current. 
The  scale  of  the  voltmeter  is  really  showing  a  deflection  which 
is  proportional  to  the  current.  An  ammeter,  if  of  the  shunt 
type,  is  really  a  sensitive  voltmeter  which  gives  a  deflection 
proportional  to  the  fall  of  potential  through  a  standard 
resistance. 

In  many  circuits  the  product  of  current  strength  and  resist- 
ance iR  is  of  great  importance.  For  convenience  it  is  called 
the  potential  drop,  or  the  iR  drop  along  the  circuit. 

104.  The  Magnetic  Effect  of  the  Current.     The  electric 
current  through  a  conductor  is  always  accompanied  by  a  mag- 


IV,  §  104]  MAGNETIC  EFFECT  141 

netic  field  in  the  region  surrounding  the  conductor.  A  mag- 
netic pole  placed  in  this  field  will  be  acted  on  by  a  force.  By 
means  of  the  action  of  this  force  on  a  magnetic  needle  the 
presence  of  a  current  in  a  conductor  can  be  ascertained. 
Since  the  force  is  proportional  to  the  current  strength,  it  also 
affords  a  direct  measure  of  the  current  strength.  This  magnetic 
force  action  does  not  depend  upon  the  kind  of  conductor,  but  is 
present  alike  with  metallic  and  with  electrolytic  conductors. 

The  deflection  of  the  magnetic  needle  when  brought  near  to 
a  conductor  through  which  current  is  flowing  was  first  ob- 
served by  Oersted  in  1819.  This  observation  was  quickly 
followed  by  the  discovery  that  the  force  action  is  at  right 
angles  to  the  conductor,  and  in  a  plane  perpendicular  to  the 
axis  of  the  conductor. 

Laplace  assumed  that  the  magnetic  field  strength  dF,  due  to 
a  current  it  through  a  short  element  of  length  ds,  is  propor- 
tional directly  to  that  length  and  to  the  strength  of  the  cur- 
rent, and  inversely  to  the  square  of  the  perpendicular  distance 
from  ds.  He  expressed  this  relation  in  the  formula 

(5)  dF=fci|i 

The  unit  of  current  strength  is  so  chosen  that  k  =  1 . 

The  relation  expressed  in  equation  (5)  cannot  be  verified 
directly  for  short  elements  of  the  conductor,  because  steady 
currents  can  only  be  thought  of  as  flowing  in  complete  cir- 
cuits. If  the  expression  for  dF  is  integrated  with  proper 
regard  for  the  geometric  form  and  the  extent  of  the  conducting 
path,  the  derived  results  are  fully  confirmed  by  experimental 
tests.  Such  tests  were  first  performed  by  Biot,  Savart,  and 
Ampere.  Indeed,  the  magnetic  field  due  to  a  long,  straight 
wire  was  established  by  Biot  and  Savart  experimentally  before 
the  general  law  of  equation  (5)  was  formulated. 

A  conductor  carrying  a  constant  current  is  to  be  thought  of 


w 

ds 


142  ELECTRIC  CURRENTS  [IV,  §  104 

as  surrounded  by  a  magnetic  field,  the  lines  of  force  being 
represented  by  concentric  circles  which  lie  in  planes  at  right 
angles  to  the  axis  of  the  conductor.  The  direction  of  these 
lines  is  clockwise  as  one  looks  along  the  conductor  in  the 
direction  in  which  the  current  flows,  and  the  strength  of  the 
field,  or  the  force  on  a  unit  pole,  is  determined  by  integrating 
the  equation  (5). 

105.  The  Magnetic  Field  due  to  a  Long  Straight  Wire. 
The  wire  WW,  Fig.  67,  is  assumed  to  be  carrying  a  current  i, 

in  a  direction  vertically  downward, 
the    magnetic    force    at    P    being 
toward   the   reader.     Let    the    per- 
pendicular  distance  from  P  to  the 
_  .  wire   be   denoted   by   r,   and    let   I 
denote  the  distance  from  P  to  the 
short   element  of   the  wire  ds.     If 
this  element  were  at  right   angles 
FlGl  67'  to  I,  the  force  dFP  at  P  would  be 

given  by  equation  (5)  ;  but,  since  the  effective  length  of  the 
element  is  ds  cos  0,  we  have  instead 

(6)  dFP  =  {-^Gose. 

The  total  force  at  P,  due  to  that  part  of  the  wire  above  0,  is 
given  by  integrating  both  sides  of  (6)  between  the  limits  zero 
and  infinity.  The  result  must  be  doubled  in  order  to  include 
the  effect  of  that  part  of  the  wire  below  0.  Remembering 
that  cos  0  =  r/l,  and  that  i  is  a  constant,  we  may  write 


(7)  ^  = 

Substituting  for  I  its  value  (r2  +  s2),  we  find 


MAGNETIC  EFFECT 


143 


IV,  §  106] 

whence  l 
(8) 


The  work  w  done  in  moving  a  unit  magnetic  pole  once 
around  a  conductor  carrying  a  current  i  is  readily  determined 
by  multiplying  the  force  as  given  in  equation  (8)  by  the 
length  of  path.  Since  the  length  of  the  circular  pathls  2  TTT, 
we  have 

W=—  .2*r  =  ±iri. 


(9) 


This   result  will   be  expressed  in  ergs  when  i  and  r  are  ex- 
pressed in  C.  G.  S.  absolute  units. 

106.  The  Magnetic  Field  Strength  at  a  Point  in  the  Axis 
of  a  Circular  Current.     Let  the  points  A  and  B,  Fig.  68, 


P       E 


FIG.  68. 


represent  the  intersections  with  the  paper  of  a  circular  loop 
of  wire,  whose  plane  is  normal  to  the  plane  of  the  paper.  If 
the  current  is  flowing  in  at  B  and  out  at  A,  the  lines  of  force 
will  be  represented  by  the  concentric  arcs.  At  P  the  force 
will  be  in  a  direction  PZ),  tangent  to  the  arc,  and  at  right 
angles  to  AP.  From  equation  (6),  §  105,  the  force  at  P  due 

1  The  integration  is  as  follows : 
ds 


144  ELECTRIC  CURRENTS  [IV,  §  106 

to  a  short  element  ds,  at  right  angles  to  the  line  AP,  is  given 
by  the  equation 

(10)  dF=—. 

The  component  dFa  of  this  force  along  the  axis  is  given  by 
the  equation 


a  -  , 

I  ILL 

or 

(11)  dFa  = 


This  being  the  force  at  P  in  the  direction  PE,  due  to  an  ele- 
ment of  length  ds,  the  total  force  at  P  in  the  same  direction 
due  to  the  entire  loop  is  given  by  integrating  both  sides  of 

(11)  around  the  circle  ;  whence  we  have  * 

(12)  F  =    27rir* 

If  the  loop  is  made  up  of  n  turns  instead  of  one,  the  total 
force  FP  at  P  in  the  direction  of  the  axis  is 

(13)  Fp 


The  component  ED  will  be  annulled  by  an  equal  component 
due  to  an  element  ds  on  the  opposite  side  of  the  loop,  and 
these  components  at  right  angles  to  the  axis  annul  one 
another  for  every  position  about  the  axis.  The  only  effective 
force  is  that  along  the  axis  as  given  by  equations  (12)  and  (13). 
If  the  point  P  is  moved  back  to  the  center  of  the  loop,  d 
becomes  zero  ;  hence,  the  force  Fc  at  the  center  is 

(14)  Fc 


r 

1  This  integration,  since  i,  r,  and  d  are  all  constant,  is  as  follows: 

.,       as  =  —  ir 
(r2-M2)f 


IV,  §  107] 


MAGNETIC  EFFECT 


145 


Fm 


107.  The    Single-coil   Tangent    Galvanometer.      Let   the 

direction  of  the  magnetic  meridian  be  represented  by  the  line 
NS,  Fig.  69.     The  points  A  and  B  represent  the  intersections 
with  the  plane  of  the  paper  of  a 
circular   coil   of   wire   of  n    turns,  ^-^ 

whose  plane   lies  in   the   magnetic  ©  ^ 

meridian.  A  magnetic  needle  ns, 
of  length  I,  is  suspended  by  a  fiber 
attached  at  o,  which  is  at  right  an- 
gles to  the  plane  of  the  paper.  The 
horizontal  component  of  the  earth's 
field  is  denoted  by  H,  and  the  mag- 
netic force  due  to  the  current  is 
denoted  by  F. 

The  magnetic  strength  of  the  pole 
of  the  needle  is  denoted  by  ra  in 
C.  G.  S.  units.  There  are  two  equal, 
oppositely  directed  forces  Fm  which 
act  on  the  two  ends  of  the  needle, 
and  tend  to  turn  it  into  a  position 
parallel  to  the  direction  of  the  field.  Moreover,  there  are  two 
forces  Hm  acting  on  the  poles,  tending  to  restore  the  needle 
to  its  position  of  equilibrium.1  The  needle  is,  therefore, 
under  the  influence  of  two  couples,  the  deflecting  couple  and 
the  restoring  couple.  When  the  moments  of  these  two  couples 
are  equal,  the  needle  will  take  up  some  definite  position, 
making  an  angle  0  with  its  original  position  in  the  magnetic 
meridian. 

Equating  the  two  moments,  we  have 


Hm, 


OB 

8 
FIG.  69. 


(15) 


Fml  cos  0  =  Hml  sin  0, 


1  The  suspension  fiber  also  supplies  a  restoring  torque,  but  this  is  assumed 
small  enough  to  be  neglected  in  all  cases  except  where  the  highest  precision 
is  required. 
L 


146  ELECTRIC   CURRENTS  [IV,  §  107 

whence,  dividing  by  ml  cos  0,  we  find 

Substituting  in  (16)  the  value  of  F  given  by  equation  (14),  we 
have 

(17)  ^  = 

whence 

(18) 


which  may  be  written  in  the  form 

(19)  i  =  /f  tan  0, 
where  K  =  Hr/(2  vri). 

This  equation  gives  the  value  of  the  current  strength  in 
terms  of  a  constant  and  the  tangent  of  the  angle  of  deflection. 
For  this  reason,  this  form  of  galvanometer  is  known  as  the 
tangent  galvanometer.  The  current  strength  is  expressed  in 
absolute  units  when  H  and  r  are  expressed  in  absolute  units. 
If  i  is  to  be  given  in  amperes  (see  §  5),  the  equation  becomes 

(20)  i  =  10  A"  tan  0. 

It  will  be  seen  that  for  any  given  instrument  r  and  n  will 
be  constant,  while  for  any  assigned  location  H  may  be  con- 
sidered constant  during  the  time  of  using  the  instrument.  In. 
any  single-coil  tangent  galvanometer  the  length  of  the  needle 
must  be  small  as  compared  to  the  diameter  of  the  coil ;  more- 
over, it  must  be  carefully  centered,  for  otherwise,  as  it  is 
deflected,  the  poles  pass  into  regions  in  which  the  field  strength 
is  not  constant. 

The  tangent  galvanometer  affords  a  ready  means  of  compar- 
ing current  strengths,  or  of  measuring  them  in  absolute  units. 
At  the  present  time,  with  high-grade,  direct-reading  instru- 
ments, potentiometers,  and  zero  methods,  it  is  difficult  to 


IV,  §  108] 


MAGNETIC  EFFECT 


147 


appreciate  the  importance  of  the  instrument  to  the  electrical 
laboratory  of  an  earlier  period.  Its  present  usefulness  lies  in 
the  illustration  of  fundamental  principles,  „., 

rather  than  in  practical  measurements. 


108.  The  Double-coil  Tangent  Gal- 
vanometer. In  the  case  of  the  single- 
coil  tangent  galvanometer  it  was  assumed 
that  the  needle  was  short,  and  that  it  did 
not  swing  out  of  a  uniform  field  at  any 
time.  By  using  two  coils  with  their 
planes  parallel,  a  much  longer  needle  may 
be  used,  the  field  is  more  nearly  uniform, 
and  a  greater  precision  may  be  attained. 
The  general  relations  of  the  two  coils 
AB  and  A'B'  to  the  magnetic  merid- 
ians NS9  and  to  the  needle  ns,  are 
shown  in  Fig.  70.  The  direction  of 
the  resultant  field  due  to  the  current  is  CD.  The  plane  of 
each  coil  is  at  a  distance  d  from  the  needle.  From  equation 
(13)  the  force  at  the  needle  due  to  one  coil  is 


(21) 


where  the  symbols  have  the  same  meanings  as  in  §  106.    Since 
there  are  two  coils  in  this  case,  the  total  force  will  be 


(22) 


F= 


An  arrangement  frequently  used  is  that  for  which  d  =  r/2. 
In  that  case,  the  equation  (22)  becomes 

4  Trnir2 
(23)  F-' 


148  ELECTRIC   CURRENTS  [IV,  §  108 

Since  F=  Ht&nO,  we  may  write 


whence 


or 

(25)  i  =  -JT  tan  0, 

where 


327m 

With  a  large  and  accurately  constructed  instrument,  r  and 
n  are  readily  determined.  It  is  with  instruments  based  upon 
extensions  of  these  principles  that  absolute  determinations  of 
current  strength  are  made. 

109.  The  Magnetic  Field  Strength  at  the  Center  of  a  Long 
Solenoid.  Figure  71  represents  a  section  through  a  long  sole- 
noid AA'/BB'9  of  length  L  cm.,  radius  r  cm.,  wound  with  wire 


n  nf 

FIG.  71. 

of  diameter  x  cm.,  and  with  n  turns  for  each  centimeter  of 
length.  It  is  desired  to  find  the  field  strength  at  a  point  P 
in  the  center  of  the  solenoid.  The  effect  of  the  n'  turns  lying 
in  the  element  of  length  act',  which  is,  for  the  present,  con- 
sidered very  short  as  compared  with  L,  is  given  by  equation 
(13)  in  the  form 

(26)  Fp  =   2vnW  . 

(y*  +  d*)4 


IV,  §  109]  MAGNETIC  EFFECT  149 

Since  n'  is  the  number  of  wire  turns  in  a  length  aa',  we  have 

aar  =  n'x, 
provided  the  wire  is  closely  wound,  or 

(27)  n'  =  —  • 

Drawing  a'c  normal  to  aP,  and  writing  the  proportions  be- 
tween corresponding  sides  of  the  similar  right  triangles  aa'c 
and  aPe,  we  have 

(28)  22._«?. 

a'c       r 

We  know  by  trigonometry  that 


When  dO  is  small,  a'P  may  be  set  equal  to  aP,  and  sin  dd  may 
be  set  equal  to  dO.     Making  these  substitutions,  (28)  becomes 


(29) 


Hence,  the  value  of  n'  in  (27)  becomes 

(30)  n'-«^». 

rx 

Putting  this  value  of  n'  in  (26),  we  find 

(31)  Ff 


Moreover,  it  is  evident  that  we  have 

(r2-f-d2)=oP2,  ^-=sin0,  -=n, 
aP  x 

where  n  is  the  number  of  turns  per  centimeter  on  the  solenoid. 
Whence  the  force  at  P  in  terms  of  the  number  of  turns  per 
centimeter  is  given  by  the  equation 

(32)  FP 


150  ELECTRIC   CURRENTS  [IV,  §  109 

To  find  the  value  of  the  field  at  P  due  to  all  the  turns 
throughout  the  entire  length  of  the  solenoid,  it  is  necessary 
simply  to  integrate  this  expression  with  respect  to  6  between 
the  limits  0  and  ?r.  Setting  H  equal  to  this  value  of  the  mag- 
netic field  strength,  we  have 


=2frni  C 


ori 
(33) 

Remembering  that  n  is  the  total  number  of  turns  divided 
by  the  length  of  the  solenoid,  equation  (33).  may  be  written  in 
the  form 

(34)  H-i.Zi. 

It  will  be  seen  that  the  limits  0  and  ?r  for  the  angle  6  cor- 
respond to  the  assumed  condition  that  L  is  very  large  compared 
to  r. 

The  value  of  H  may  be  expressed  either  in  dynes  per  unit 
pole  or  in  lines  per  square  centimeter.  In  case  the  current  is 
measured  in  amperes,  the  equation  (33)  becomes 

(35)  •  *  =  ^***- 

Since  this  is  the  magnetic  field  strength  or  flux  density  at  the 
center,  the  total  flux  <f>  through  the  solenoid  is  given  by  the 
equation 

(36)  *-J5""*4        . 

where  A  is  the  area  of  cross-section  of  the  coils.  It  is  here 
assumed  that  the  magnetic  field  is  uniform  over  the  entire 
area  of  the  solenoid.  When,  for  any  reason,  a  magnetic  field 

1  This  integration  is  as  follows : 

f  *  sin  6  d6  =  f  —  cos  01 "  =  1  + 1  =  2. 


IV,  §  110]       THE  ELECTRODYNAMOMETER 


151 


of  known  strength  is  required,  it  is  most  frequently  realized 
by  means  of  the  long  solenoid  carrying  a  known  current. 

The  magnetic  field  is  uniform  for  a  certain  region  near  the 
center  of  the  solenoid,  but  toward  the  ends  it  is  no  longer 
parallel  to  the  axis,  and  its  value  is  not  given  by  equation  (33). 
If  the  long  solenoid  is  bent  into  a  circular  form  with  the  ends 
joined,  thus  forming  a  toroidal  coil,  the  end  effects  and  the 
external  field  vanish,  and  the  lines  of  force  are  circles  with  their 
centers  lying  on  the  axis  of  the  tore.  If  the  wire  turns  are  close 
together,  the  windings  may  be  considered  as  forming  approxi- 
mately a  uniform  current  sheet,  within  which  the  value  of  the 
uniform  magnetic  field  is  given  by  equations  (33)-(35). 

110.  The  Electrodynamometer.      The  electrodynamometer 

is  an  instrument  of  great  utility  for  the  measurement  of  cur- 
rent strength,  voltage,  or  power,  in  either  direct- 
current  or  alternating-current  circuits.     For  the 
present  it  will  be  treated  as  a  current-measuring 
device. 

It  consists  essentially  of  two  rectangular  coils, 
connected  in  series,  placed  with  their  planes  ver- 
tical and  at  right  angles  to  one  another,  as  shown 
in  Fig.  72.  One  coil  is  fixed  in  position  while  the 
other  is  hung  from  a  torsion  head  by  a  light  fiber 
of  silk,  so  that  it  is  free  to  rotate  about  a  vertical 
axis.  A  light  spiral  spring  surrounds  the  sus- 
pending fiber,  and  is  attached  to  the  movable  coil 
and  to  the  torsion  head.  This  spring  furnishes  the  control  for 
the  suspended  system.  Electrical  connection  with  the  movable 
coil  is  provided  by  means  of  mercury  cups,  into  which  its  ter- 
minals dip,  and  which  are  placed  directly  in  a  vertical  line 
below  the  point  of  suspension.  Some  clamping  arrangement 
is  usually  provided  to  prevent  damage  to  the  suspended  system 
during  transportation. 


FIG.  72. 


152  ELECTRIC   CURRENTS  [IV,  §  110 

When  current  is  passed  through  the  two  coils  connected  in 
series,  their  magnetic  fields  react  to  produce  a  torque,  which 
rotates  the  movable  coil  about  its  vertical  axis.  This  torque 
may  be  opposed  by  twisting  the  torsion  head  and  the  attached 
spring  through  a  certain  angle,  until  the  torsion  of  the  spring- 
just  compensates  the  torque  due  to  the  reacting  fields.  This 
angle  is  read  by  means  of  a  pointer  attached  to  the  head,  which 
plays  over  a  graduated  scale  on  the  top  of  the  frame  of  the 
instrument. 

The  torque  due  to  the  reacting  fields  is  proportional  to*  the 
current  strength  in  each  coil,  and  hence,  to  the  square  of  the 
current  strength.  The  compensating  torque  of  the  spring  is 
directly  proportional  to  the  angle  through  which  the  torsion 
head  is  rotated  in  order  to  keep  the  suspended  coil  in  its  posi- 
tion of  equilibrium.  It  will  be  seen  that  the  direction  of  the 
deflecting  torque  is  not  changed,  even  though  the  current  is 
reversed  through  the  coils. 

From  these  considerations,  it  will  be  seen  that 


where  <f>  is  the  angle  through  which  the  torsion  head  is  rotated 
in  order  to  maintain  the  movable  coil  in  its  zero  position,  or 
position  of  equilibrium,  and  A:'  is  a  constant  which  depends  on 
the  stiffness  of  the  spring,  the  dimensions  of  the  coils,  and  the 
number  of  wire  turns.  The  preceding  equation  may  be  written 
in  the  form 


(37)  i 

where  k  =  -\/k'.     If  the  value  of  k  is  known,  the  value  of  i 
may  be  readily  computed. 

Thus,  it  appears  that  the  square  root  of  the  observed  angle 
through  which  the  spring  is  rotated,  multiplied  by  a  constant, 
gives  the  value  of  the  current  strength.  This  constant  may  be 
determined  by  passing  a  current  of  known  strength  through 


IV,  §  110]       THE  ELECTRODYNAMOMETER  153 

the  instrument  and  observing  the  angle  through  which  the. 
torsion  head  must  be  rotated  in  order  to  keep  the  moving  coil 
in  its  initial  position  of  equilibrium.  The  current  may  be 
measured  by  any  desired  method  of  suitable  accuracy.  Since 
a  precise  determination  is  necessary  for  the  calibration  of  the 
instrument,  it  is  customary  to  use  a  silver  or  copper  voltame- 
ter in  series  with  it,  computing  the  current  strength  from 
the  gain  in  mass  of  the  cathode  in  a  measured  interval  of  time. 
If  currents  and  angles  of  twist  are  plotted  on  squared  paper, 
the  curve  will  be  parabolic. 

In  measuring  direct  currents  the  reaction  between  the  field 
of  the  movable  coil  and  the  earth's  field  may  be  considerable. 
To  avoid  error  from  this  cause,  the  instrument  is  so  placed 
that  the  plane  of  the  movable  coil  is  at  right  angles  to  the 
magnetic  meridian. 

The  electrodynamometer  may  be  calibrated  and  used  also 
for  the  measurement  of  voltage  and  power  in  either  direct  or 
alternating-current  circuits.  It  is  sometimes  equipped  with  a 
mirror  and  scale,  and  the  deflection  angles  are  read  directly 
instead  of  being  annulled  by  a  torsion  spring.  When  so  used, 
it  is  called  a  reflecting  electrodynamometer. 

As  a  voltmeter  this  instrument  is  made  with  many  turns  of 
fine  wire  in  the  coils,  which  are  in  series,  and  usually  with  a 
high  non-inductive  resistance  also  in  series.  This  acts  as  a 
multiplier  and  enables  the  range  of  the  instrument  to  be 
increased. 

As  a  wattmeter,  the  fixed  coil  (current  coil)  consists  of  a  few 
turns  of  large  wire,  and  its  terminals  are  connected  in  series 
with  the  circuit  in  which  the  power  consumption  is  to  be 
measured.  The  movable  coil  (pressure  coil)  is  made  of  many 
turns  of  fine  wire  in  order  to  secure  a  high  resistance,  and  is 
connected  in  parallel  with  the  power  circuit.  A  non-inductive 
resistance  is  frequently  connected  in  series  with  the  pressure 
coil, 


154  ELECTRIC  CURRENTS  [IV,  §  111 

111.  The  Heating  Effect  of  the  Current.  The  potential 
difference  between  two  points  is  measured  in  terms  of  the 
work  done  in  conveying  the  unit  charge  between  these  points. 
This  relation  is  given  by  the  equation 


or 

(38)  W=  FQ, 

where  V  is  the  potential  difference  and  W  is  the  work  done 
in  conveying  the  charge  Q.  By  setting  Q  =  it,  equation  (38) 
becomes 

(39)  W=  Vit, 

or,  if  the  value  of  V  is  substituted  from  Ohm's  law, 

(40)  W=  Vit  =  PRt. 

The  thermal  equivalent  of  the  work  is  given  by 

(41)  W=  JH, 

where  J  is  the  mechanical  equivalent  of  heat,  or  the  number 
of  work  units  equivalent  to  one  heat  unit.  Combining  (40) 
and  (41),  we  find, 

(42)  W=JH=Vit  =  i*Rt. 

The  work  will  be  given  in  ergs  when  the  electric  units  are 
all  taken  in  the  absolute  C.  G.  S  system,  and  in  joules  when 
the  volt,  ampere,  ohm,  and  second  are  used.  With  the  abso- 
lute units  J  has  the  approximate  value 

J=  4.18  X  107  ergs  per  calorie  ; 
while,  with  the  practical  units, 

J=  4.18  joules  per  calorie. 

If  both  sides  of  equation  (42)  are  divided  by  the  time,  the 
power  relations  of  the  electric  quantities  are  obtained  in  the 
form 


IV,  §  111] 


(43) 


HEATING  EFFECT 


155 


==         = 

t         t 


which  expresses  P  in  watts  when  practical  units  are  used  con- 
sistently. 

The  equations  (42)  and  (43)  might  be  used  to  find  any  one 
of  the  electric  quantities  involved,  all  the  others  being 
known.  There  are  other  more  accurate  methods  available  for 
measuring  current  and  resistance,  however,  and  the  equations 
are  more  frequently  used  to  determine  the  mechanical  equiva- 
lent of  heat,  J. 

The  method  here  described  is  that  of  the  flow  calorimeter. 
Figure  73  shows  the  arrangement  of  the  parts.  A  spiral  coil 


FIG.  73. 

of  wire  S  within  a  glass  tube  carries  a  current  which  is 
measured  by  the  ammeter  Am.  A  voltmeter  Vm  placed  across 
the  terminals  of  the  coil  gives  the  potential  difference  between 
its  ends.  From  these  readings  the  rate  of  energy  supply  to 
the  coil  is  found.  This  energy  heats  the  wire.  If  a  stream  of 
water  is  made  to  flow  continuously  through  the  tube,  there 
will  be  a  constant  difference  in  the  temperature  of  inflow  and 
outflow,  provided  the  rate  of  energy  supply  by  the  current  is 
just  equal  to  the  rate  of  energy  withdrawn  by  the  water  stream. 
From  equation  (42)  we  may  write 


or 


H 


156  ELECTRIC   CURRENTS  [IV,  §  111 

The  heat  removed  by  the  water  stream  is  given  by  the  product 
of  the  mass  of  water  and  the  difference  between  the  initial 
and  final  temperatures,  whence  we  have 

(44)  J=—™- 

ro  («,-«,) 

In  the  continuous-flow  calorimeter,  heat  is  carried  away  at 
a  uniform  rate,  being  absorbed  by  the  water  which  flows 
steadily  through  the  tube.  If  the  temperature  of  the  water 
supply  is  constant,  and  if  the  flow  is  maintained  at  a  uniform 
rate,  the  thermal  condition  will  become  fixed,  as  will  the  elec- 
tric condition.  That  is,  the  temperatures  of  inflow  and  out- 
flow will  become  constant,  and  the  resistance  of  the  wire  will 
not  change.  Under  these  conditions  there  are  no  corrections 
to  be  made  for  the  thermal  capacity  of  the  apparatus.  By 
keeping  the  flow  of  water  steady,  and  the  mean  temperature 
of  inflow  and  outflow  within  five  degrees  of  the  room  tempera- 
ture, corrections  for  radiation  and  conduction  become  very 
small  and  may  be  neglected. 

112.  Laboratory  Exercise  XXV.  To  determine  the  mechani- 
cal equivalent  of  heat  with  the  flow  calorimeter. 

APPARATUS.  Flow  calorimeter  with  accessories,  ammeter, 
voltmeter,  control  rheostat,  two  thermometers,  watch,  and 
source  of  steady  current. 

PROCEDURE.  (1)  Arrange  the  apparatus  as  shown  in  Fig.  73. 
Adjust  the  flow  of  water  until  it  is  steady,  with  a  difference 
in  temperature  between  inflow  and  outflow  of  from  three  to 
five  degrees.  Let  the  water  flow  for  a  few  minutes  before 
taking  readings  so  that  the  temperatures  may  become  con- 
stant. The  thermometer  readings  should  be  estimated  to  one 
hundredth  of  a  degree,  and  the  two  instruments  should  be 
compared  before  beginning  the  experiment. 

(2)  See  that  no  air  bubbles  are  lodged  on  the  wire  turns  of 
the  coil.  After  the  current  has  flowed  a  few  minutes,  set  a 


IV,  §  113] 


ELECTROLYTIC  EFFECT 


157 


weighed  vessel  in  position  to  receive  the  outflowing  water, 
and  note  the  exact  time  at  which  the  flow  into  the  vessel 
begins.  Take  simultaneous  readings  of  the  ammeter,  volt- 
meter, and  both  thermometers  at  half-minute  intervals,  re- 
cording these  values  in  a  table  previously  ruled.  When 
one  or  two  liters  of  water  has  passed,  remove  the  vessel, 
note  the  exact  time,  and  record  it.  Weigh  the  water  col- 
lected, and  compute  the  number  of  calories  of  heat  absorbed 
by  the  water. 

(3)  Repeat  for  five  sets  of  observations,  using  different  rates 
of  flow  and  different  values  of  the  current.  Let  the  final  re- 
sult be  the  mean  of  the  five  thus  found.  The  data  may  be 
arranged  as  shown  in  the  following  table : 


TIME 

i 

V 

TEMP. 
IN 

TEMP. 
OUT 

MASS  OF 
WATEK 

J 

113.  The  Electrolytic  Effect  of  the  Current.  If  a  wire 
carrying  a  current  of  electricity  is  cut  and  its  ends  are  sub- 
merged in  a  jar  containing  a  water  solution  of  an  inorganic 
acid  or  salt,  current  will  still  continue  to  flow,  and  there  will 
be  a  deposit  of  ions  on  the  cathode,  that  is,  the  terminal  from 
which  current  leaves  the  solution.  The  mass  M  of  this  de- 
posit is,  from  Faraday's  laws,  proportional  to  the  amount  of 
charge  Q  passing.  This  may  be  expressed  by  the  formula 

M=zQ, 

where  z  is  the  mass  deposited   by  the  unit  of  charge.     We 
have  also 


158 


ELECTRIC   CURRENTS 


[IV,  §  113 


hence,  if  the  current  strength  is  constant  throughout  the  time 
tt  its  value  may  be  found  from  the  relation 


(45) 


zt 


FIG.  74. 


The  value  of  z,  which  is  called  the  electrochemical  equiva- 
lent, is  characteristic  of  the  substance  deposited.  It  has  been 
so  accurately  determined  for  silver  and  copper  that  voltameters 
containing  solutions  of  these  metals  are 
used  for  precise  measurements  of  current 
strength.  The  method  is  valuable  for 
the  calibration  of  current-measuring  in- 
struments, but  it  is  not  commonly  used 
outside  of  the  precision  laboratory.  It 
is  slow  and  requires  considerable  equip- 
ment, and  it  gives  results  that  are  more 
accurate  than  are  required  in  practice. 

The  silver  voltameter  is  used  for  the 
most  precise  determinations,  and  its  form 
is  usually  that  shown  in  Fig.  74.  The  cathode  S  is  a  plate  of 
pure  silver  so  mounted  that  it  can  be  immersed  in  a  solution 
of  silver  nitrate  contained  in  a  platinum  bowl  P.  The  inter- 
national ampere  is  denned  in  terms  of  the  silver  voltameter ; 
the  value  of  z  for  silver  being  0.0011180 
gram  per  coulomb.  (See  §  6.) 

The  copper  voltameter  is  easier  to 
use  than  the  silver  voltameter,  and  is 
nearly  as  precise  in  its  results.  It  usu- 
ally takes  the  form  shown  in  Pig.  75. 
The  middle  plate  is  the  cathode  and 
the  two  outside  plates  are  joined  and  constitute  the  anode.  A 
twenty-five  per  cent  solution  of  copper  sulphate  with  the  addi- 
tion of  one  or  two  per  cent  of  sulphuric  acid  is  used.  The 
electrochemical  equivalent  of  copper  is  0.0003294  gram  per 


FIG. 


IV,  §  114] 


ELECTROLYTIC  EFFECT 


159 


coulomb.     This  value  varies  slightly  with  the  current  density 
and  with  the  temperature  and  concentration  of  the  solution. 

114.  Laboratory  Exercise  XXVI.  To  determine  the  constant 
of  an  electrodynamometer  with  the  copper  voltameter. 

APPARATUS.  Electrodynamometer,  copper  voltameter  in 
duplicate  with  accessories,  rheostat,  and  reversing  switch. 

PROCEDURE.  (1)  Set  up  the  instrument  so  that  the  coils 
are  at  right  angles  to  one  another,  and  with  the  plane  of  the 
movable  coil  at  right  angles  to  the  magnetic  meridian.  Adjust 
the  leveling  screws  until  the  coil  swings  freely,  and  set  the 
torsion  head  against  its  stop  on  zero.  Bead  accurately  the 


FIG.  76. 

position  of  the  coil  pointer,  and  take  this  as  the  zero  or  equi- 
librium position.     Stops  are  provided  to  limit  the  deflection. 

(2)  Connect  the  circuit  as  in  Fig.  76,  using  a  twisted  pair  of 
wires  to  the  electrodynamometer.     Pass  a  current  of  suitable 
strength  through  the  circuit  and  note  the  angle  of  compensa- 
tion.    Eeverse  the  current  and  note  whether  the  compensation 
angle  varies.     Do  this  for  two  positions,  with  the  plane  of  the 
movable  coil   respectively  parallel  with  and  perpendicular  to 
the  magnetic  meridian.     Any  readings  made  should  be  with 
the  former  position,  and  with  reversed  current. 

(3)  Adjust  the  current  to  a  suitable  value  and  proceed  with 
the  voltameter  determination  as  outlined  in  §  113.     Let  the 
current  pass  for  at  least  half  an  hour,  reversing  every  two 
minutes,  and  record  the  compensation  angles. 

(4)  Calculate  the  current  strength  from  equation  (45),  and 
find  the  value  of  the  constant  in  equation  (37). 


160  ELECTRIC  CURRENTS  [IV,  §  114 


EXERCISES 

1.  A  circular  loop  of  wire  of  radius  60  cm.  is  placed  with  its  plane 
vertical  and  in  the  magnetic  meridian.     A  current  of  10  amperes  flows 
north  at  the  top  of  the  coil.     A  south  magnetic  pole  of  200  units  strength 
is  placed  on  the  axis  of  the  coil,  at  a  distance  of  80  cm.  from  its  plane. 
Calculate  the  force  on  this  pole.     Show  clearly  in  a  diagram  its  direction. 
What  is  the  force  on  the  pole  if  placed  at  the  center  of  the  loop  ?     What 
is  its  direction  ? 

2.  Make  a  diagram  which  will  show  clearly  the  magnetic  field  reactions 
in  the  electrodynamometer.     In  the  case  of  the  error  due  to  the  earth's 
field,  which  one  of  the  three,  (a)  total  force,  (6)  horizontal  component, 
(c)  vertical  component,  is  the  effective  one.     Make  a  diagram  showing 
clearly  the  directions  of  the  reacting  fields  and  of  the  resulting  forces  and 
torques.      Assume  approximate   dimensions,  and  calculate  the  possible 
value  of  the  torque  due  to  this  cause. 


CHAPTER   V 
CAPACITY  AND   THE  CONDENSER 

PART  I.     DEFINITIONS  AND  UNITS 

115.  Fundamental  Ideas  and  Definitions.  Up  to  this  time 
we  have  considered  electricity  as  resembling  in  some  ways  an 
incompressible  fluid,  and  we  have  assumed  that  all  parts  of 
the  circuit  carried  the  same  current  strength  at  the  same  time. 
In  this  and  the  following  chapter  new  aspects  of  electric 
circuits  will  be  presented,  in  which  there  will  be  considered 
the  storage  of  energy  in  certain  parts  of  the  circuit. 

If  the  wire  connecting  the  poles  of  a  battery  is  cut,  its  ends 
will  be  at  a  definite  difference  of  potential,  and  the  charges 
residing  on  the  free  ends  will  be  small.  If,  however,  these 
free  ends  of  the  wires  are  expanded  into  plates  with  large 
surface  areas,  there  will  be  a  momentary  current  through  the 
circuit,  and  a  greater  charge  will  accumulate  on  the  plates. 
As  the  plate  area  is  increased,  and  as  the  distance  between  the 
plates  is  made  less,  the  charge  on  the  plates,  for  the  same 
potential  difference,  increases.  This  ability  of  the  system  of 
conductors  to  hold  or  store  a  quantity  of  electricity  is  called 
the  capacity1  of  the  system.  Such  a  system  of  conducting 
plates  is  called  a  condenser.  It  is  a  device  by  means  of  which 
the  capacity  of  an  isolated  conductor  can  be  very  greatly  in- 
creased, due  to  the  presence  near  it  of  another  charged  con- 
ductor. This  other  conductor  may  be  connected  to  the  earth  or, 
more  commonly,  to  the  opposite  pole  of  the  electric  generator. 

1  In  order  to  distinguish  electrostatic  capacity  from  current-carrying  ca- 
pacity, etc.,  the  term  capacitance  is  sometimes  used. 
M  161 


162  CAPACITY  AND  THE  CONDENSER     [V,  §  115 

The  capacity  of  a  condenser  can  be  shown  to  be  directly 
proportional  to  the  area  of  surface  of  its  plates,  and  inversely 
to  the  distance  between  them.  In  order  to  realize  the  greatest 
possible  capacity  in  a  small  space,  a  great  many  very  thin 
sheets  of  metal  foil  are  used  for  the  conductors,  and  these  are 
separated  by  selected  sheets  of  thin  mica  in  the  higher  grades 
of  condensers,  or  by  sheets  of  paraffined  paper  in  the  cheaper 
grades. 

116.  Classification  of  Condensers.     Condensers   may  be 
grouped  in  two  classes.      One  class  is  that  in  which  the  di- 
electric must  sustain  a  high  potential.     Such  a  condenser  con- 
sists of  a  few  plates  widely  separated,  and  the  capacity  is  too 
small  to  measure  by  the  ordinary  methods.     Such  condensers 
are    commonly   used    in    high-frequency,    alternating-current 
circuits.     Their  properties  and  the  methods  of  making  meas- 
urements with  them  are  treated  in  the  larger  works  on  the 
theory  and  equipment  of  wireless  telegraphy. 

Condensers  of  the  other  class  have  larger  capacity  and  are 
intended  for  use  with  low-voltage  batteries  and  ordinary  gal- 
vanometers. They  have  many  layers  of  thin  foil  separated  by 
thin  dielectric,  closely  pressed  together. 

117.  Units  of  Capacity.     In  Fig.  77,  AB  represents  such  a 
system  of  interleaved  plates.     If  the  key  K  is  pressed  to  b, 

a  potential  difference  V  is  applied 
to  the  terminals  of  the  condenser, 
and  a  transient  deflection  of  the 
galvanometer  is  observed.  The  zero 
position  is  quickly  regained,  how- 
ever. This  sudden  throw  of  the 

galvanometer    signifies    a    rush    of 
FIG.  77. 

electricity  into  the  condenser.     The 

condenser  is  then  said  to  be  charged.     If  the  key  K  is  then 
raised  to  a,  thus  removing  the  charging  potential  difference 


V,  §  117]  DEFINITIONS  AND  UNITS  163 

and  closing  the  condenser  circuit  through  the  galvanometer,  a 
transient  deflection  is  again  observed,  this  time  in  a  direction 
opposite  to  the  first  one.  The  condenser  is  now  discharged, 
the  plates  having  been  brought  to  the  same  potential. 

The  current  strength  l  during  charge  or  discharge  is  not  con- 
stant, as  will  be  shown  in  §§  125  and  127,  if  a  condenser  is 
included  in  the  system.  It  is  necessary  to  take  into  account 
the  total  quantity  of  charge  which  passes  rather  than  the 
current  itself.  This  is  given  by  the  expression 

(1)  Q 

where  i  is  the  instantaneous  value  of  the  current  strength. 
The  quantity  of  electricity  stored  in  a  perfect  condenser,  that 
is,  one  whose  dielectric  has  infinite  resistance  and  no  absorp- 
tion,2 is  always  found  to  be  directly  proportional  to  the  charg- 
ing potential  difference.  This  relation  may  be  written  in  the 
form 


(2)  Q 

where  C  is  a  constant  whose  value  is  given  by  the  ratio 

(3)  -      :~:  .     -  C  =  ^.  ,        '  •          : 

This  constant,  which  is  characteristic  of  the  particular  con- 
denser, is  the  measure  of  the  capacity  of  the  condenser.  From 
equation  (3)  it  is  seen  that  the  capacity  is  numerically  equal 
to  the  charge  in  the  condenser  when  unit  potential  difference 
is  impressed  across  its  terminals. 

If  Q  and  V  are  given  in  absolute  C.  G.  S.  electromagnetic 
units,  C  will  be  expressed  in  the  same  system.     A  condenser 

1  In  cables  and  transmission  lines  the  capacity  is  distributed,  and  the  cal- 
culation of  the  current  strength  at  any  time,  and  at  any  point  along  the  con- 
ductor, becomes  somewhat  complicated.    The  theory  of  problems  of  this  class 
is  given  in  The  Propagation  of  Electric  Currents  in  Telegraph  and  Telephone 
Conductors  by  FLEMING  (Van  Nostrand,  1911). 

2  See  §  119. 


164  CAPACITY  AND   THE   CONDENSER      [V,  §  117 

will  have  unit  capacity  when  unit  potential  difference  develops 
in  it  the  unit  charge.  In  order  to  express  capacity  in  practical 
units,  Q  must  be  given  in  coulombs  and  "Fin  volts,  whence 

/4\  [Qc.G.S.   X  10  ]  coulombs  _  r  fl  v   1  A9H 

w  -fv  --  v      -8         -L^c.G.s.  x  iujf 

L  r  c.G.S.  X 


Any  given  charge  of  Q  absolute  units  will  be  represented  by 
a  number  ten  times  as  great  when  expressed  in  coulombs  ; 
and  any  given  number  of  absolute  units  of  potential  difference 
will  be  divided  by  10s  in  order  to  give  the  equivalent  number 
of  volts.  The  number  of  absolute  capacity  units  must  then  be 
multiplied  by  109  in  order  to  give  the  equivalent  number  of 
practical  units.  This  means  that  the  value  of  the  absolute 
unit  of  capacity  is  109  times  as  great  as  the  practical  unit  which 
corresponds  to  the  volt  and  the  coulomb.  This  practical  unit 
of  capacity  is  called  the  farad,  and  is  the  capacity  of  a  con- 
denser which  has  a  potential  difference  of  one  volt  at  its  ter- 
minals, when  charged  with  one  coulomb  of  electricity. 

The  farad  itself  is  too  large  a  unit  to  be  useful,  being  of  an 
order  of  magnitude  much  greater  than  that  of  the  capacities 
commonly  met  in  practice,  hence,  the  millionth  part  of  the 
farad,  the  microfarad,  equivalent  to  10~15  in  absolute  units,  is 
chosen  as  a  more  convenient  and  more  practical  unit.  A  con- 
denser of  capacity  one  farad  would  be  too  enormous  to  con- 
struct :  the  height  of  a  pile  of  condenser  plates  each  one  meter 
square,  which  would  be  required  for  a  capacity  of  one  farad, 
provided  that  the  thickness  of  each  conducting  sheet  together 
with  its  mica  dielectric  were  one  millimeter,  would  be  of  the 
order  of  one  hundred  miles. 

A  submarine  cable  is  a  condenser  in  which  the  copper  core 
constitutes  one  plate,  with  the  water  as  the  other  plate,  while 
the  insulating  material  surrounding  the  core  is  the  dielectric. 
Similarly,  in  a  telephone  cable,  any  single  conductor  may  be 
regarded  as  one  plate  of  a  condenser,  the  other  plate  being  the 


V,  §  118] 


DEFINITIONS  AND  UNITS 


165 


adjacent  wire  of  a  pair,  or  the  lead  sheath  of  the  cable  itself. 
The  capacity  of  a  telephone  cable  should  not  be  greater  than 
0.08  microfarad  per  mile.  The  earth,  considered  as  an  isolated 
conductor,  has  a  capacity  of  about  700  microfarads.  The 
capacity  of  three  miles  of  average  submarine  cable  is  about 
one  microfarad.  A  laboratory  standard  frequently  used  is  one 
having  a  third  of  a  microfarad  capacity,  equivalent  to  about 
one  mile  of  cable.  The  capacity  of  a  pair  of  number  eight 
copper  wires  1000  feet  in  length  and  twelve  inches  apart  is 
about  0.0032  microfarad. 


FIG.  78. 


118.  Standards  of  Capacity.  Standard  condensers,  or  ca- 
pacity boxes  for  use  in  the  laboratory,  may  be  arranged  with 
single  values,  commonly  •£,  -J,  or  1  microfarad  in  each  box,  or 
they  may  be  subdivided, 
with  a  maximum  value  of 
one  or  more  microfarads. 
Subdivided  condensers 
are  so  arranged  in  sec- 
tions that  different  values 
of  the  capacity,  from  a 
few  hundredths  of  a  microfarad  to  the  maximum,  can  be  secured 
by  the  adjustment  of  plugs  or  switches.  Two  methods  of  con- 
necting the  separate  sections  in  subdivided  capacity  boxes  are 

shown  in  Figs.  78 
and  79.  The  ar- 
rangement shown  in 
Fig.  78  permits  of 
multiple  combina- 
tions only,  while 
with  that  shown  in 
Fig.  79,  both  series  and  multiple  combinations  are  possible. 
In  certain  geometric  forms,  notably  the  sphere,  cylinder,  and 
parallel  plate,  it  is  possible  to  calculate  the  capacity  from  the 


FIG.  79. 


166  CAPACITY  AND   THE  CONDENSER     [V,  §  118 

dimensions.  These  forms  serve  as  reliable  standards  when 
dry  air  is  used  as  the  dielectric,  but  the  capacities  will  be 
small  unless  inconveniently  large  dimensions  are  assumed.  If 
dry  air  or  vacuum  constitutes  the  dielectric  of  a  condenser, 
the  value  of  the  capacity  will  not  be  dependent  upon  the 
charging  potential,  nor  upon  the  time  for  which  it  is  applied. 
With  such  solid  dielectrics  as  glass  or  paraffin,  however,  the 
capacity  is  found  to  depend  on  the  mode  of  charging.  The 
phenomena  of  leakage,  absorption,  and  residual  charge  must 
be  taken  into  account  carefully. 

119.  Leakage,  Absorption,  and  Residual  Charge.  When 
the  dielectric  of  a  condenser  shows  a  true  conductivity,  it  is 
said  to  possess  leakage.  No  substance  can  be  regarded  as  an 
absolute  non-conductor,  though  a  pair  of  charged  plates  with 
dry  air  as  the  dielectric  will  retain  the  charge  almost  indefi- 
nitely. A  high-grade  mica  condenser  will  retain  its  charge 
for  some  hours  with  but  slight  change,  while  an  average 
paraffined-paper  condenser  shows  a  marked  falling  off  in  its 
charge  within  a  few  seconds.  This  dielectric  conductivity 
may  be  strictly  like  that  in  metals,  or  it  may  be  electrolytic  in 
type.  The  conductivity  of  dielectrics  usually  increases  with 
rise  in  temperature,  and  with  an  increase  in  the  impressed 
voltage.  Solid  substances  which  are  not  changed  in  chemical 
composition  at  high  temperatures,  such  as  glass  or  porcelain, 
become  good  conductors  when  raised  to  incandescence. 

When  a  given  potential  difference  applied  to  a  condenser 
gives  it  a  certain  charge  for  a  short  time  of  application,  and 
a  greater  charge  for  a  longer  time,  the  condenser  is  said  to 
possess  absorption.  If  such  a  condenser  with  negligible  leak- 
age is  charged  by  a  given  potential  difference  and  then  is  left 
to  itself  after  removing  the  charging  voltage,  the  potential 
difference  across  its  terminals  is  found  to  diminish  somewhat, 
at  first  rapidly,  then  slowly.  Under  the  action  of  the  charg- 


V,  §  119]  DEFINITIONS  AND  UNITS  167 

ing  potential  some  molecular  changes  probably  occur  in  the 
dielectric,  which  require  time,  and  this  strained  condition  also 
requires  time  for  recovery.  Absorption  is  by  some  writers 
called  soakage,  both  terms  arising  from  the  early  view  that 
the  electric  current  was  of  the  nature  of  fluid  flow,  and  that 
more  or  less  penetration  into  the  substance  of  the  dielectric 
occurred.  In  any  event  the  process  of  charging  affects  the 
dielectric  like  a  mechanical  stress,  and  there  are  but  few  solid 
substances  which  recover  immediately  after  the  removal  of 
such  stress.  This  is  shown  by  the  intimate  relation  which 
exists  between  the  phenomena  of  absorption  and  the  elastic 
after-effect  of  the  dielectric  substance.  With  glass,  absorption 
and  elastic  after-effect  are  both  large,  while  with  quartz 
they  are  both  practically  zero.  An  air  condenser  shows  no 
absorption. 

When  the  terminals  of  a  charged  condenser  are  connected 
by  a  conductor,  they  are  brought  to  the  same  potential,  and  the 
condenser  is  said  to  be  discharged.  If  they  are  again  con- 
nected after  the  lapse  of  a  brief  time,  another  smaller  dis- 
charge occurs,  and  this  may  be  repeated  several  times.  This 
so-called  residual  charge  is  closely  associated  with  the  absorp- 
tion of  the  dielectric,  and  is  due  to  the  slow  recovery  of  the 
dielectric  from  the  electrostatic  strain.  Condensers  with 
quartz  or  dry  air  as  the  dielectric  do  not  show  residual  charge. 

The  construction  and  use  of  a  condenser  would  be  simplified 
if  the  dielectric  material  was  free  from  the  properties  of  leak- 
age and  absorption.  The  significance  of  the  capacity  of  a  con- 
denser is  not  definite  unless  the  circumstances  of  charging  and 
discharging  are  fully  specified.  The  precision  condenser  must 
be  carefully  studied  in  order  to  ascertain  the  influence  of  tem- 
perature changes. 

Cables  and  transmission  lines  act  as  condensers.  When 
they  are  subjected  to  high  potential  differences,  the  absorption 
in  the  dielectric  may  result  in  large  residual  charges.  When  a 


168  CAPACITY  AND  THE   CONDENSER     [V,  §  119 

high-tension  circuit  is  opened,  such  condensers  should  always 
be  effectively  discharged  by  repeated  or  continuous  grounding 
before  they  are  touched,  otherwise  surprising  discharge  vol- 
tages may  develop. 

120.  Specific  Inductive  Capacity.     The  capacity  of  a  con- 
denser depends  not  only  upon  the  form  and  dimensions  of  the 
plates,  but  also  upon   the   nature  of   the   dielectric   medium. 
Suppose   the   capacity  of  a  given  condenser  with  air  as  the 
dielectric  is  <7a,  while  the  capacity  of  the  same  condenser  witli 
some  other  substance  as  the  dielectric  is  CQ.     The  specific  in- 
ductive capacity,  or  the  dielectric  constant  of  the  substance,  is 
defined  by  the  equation 

(5)  fc  =  S. 

Strictly  speaking,  the  reference  medium  for  which  the  dielec- 
tric constant  is  taken  as  unity  should  be  a  vacuum.  However, 
dry  air  differs  so  little  from  a  vacuum  in  this  respect  that  its 
dielectric  constant  may  also  be  taken  as  unity. 

Measurements  of  specific  inductive  capacity  yield  results 
which  vary  widely  with  the  physical  state  of  the  substances 
and  with  the  conditions  of  the  test.  Average  values  for  a  few 
substances  are  given  in  the  following  table : 

Petroleum 2.0 

Ebonite 2.0-3.0 

Paraffin 2.3 

Glass 2.0-10.0 

Mica       5.0-7.0 

121.  Dielectric  Strength.     When  condensers  are  to  be  used 
with  high  voltages,  the  property  of  dielectric  strength  is  quite 
as  important  as  good  insulation.     If  the  potential  difference 
impressed  exceeds  a  certain  critical  value,  the  dielectric  will 
be  pierced.     In  case  the  dielectric  is  a  gas  or  a  liquid,  its  con- 
tinuity is  restored  immediately  after  the  spark.     In  a   solid 


V,  §  122]  DEFINITIONS  AND   UNITS  169 

dielectric,  however,  the  path  of  the  spark  is  a  permanent  de- 
fect, and  if  sufficient  electric  energy  is  supplied  by  the  gener- 
ator, current  will  continue  to  flow  along  this  path  in  the  form 
of  an  electric  arc. 

The  dielectric  strength  is  expressed  in  terms  of  the  poten- 
tial difference  in  volts  (or  in  kilovolts)  required  to  pierce  a 
given  thickness  of  the  substance.  It  is  not  a  quantity  that 
can  be  very  definitely  measured.  The  results  vary  with  the 
character  of  the  voltage,  whether  direct  or  alternating,  and 
also  with  the  distance  between  the  plates  or  electrodes,  the 
shape  of  the  plates,  and  the  time  during  which  the  voltage  is 
applied.  Some  approximate  values  for  average  samples  of 
common  materials  are: 

Mica 60,000  volts  per  mm. 

Vulcanized  rubber 10,000  volts  per  mm. 

Insulating  oils 5000-10,000  volts  per  mm. 

In  any  case,  the  presence  of  moisture  greatly  lessens  the  di- 
electric strength.  Although  air  is  an  excellent  insulator,  its 
dielectric  strength  is  lower  than  that  of  most  solid  or  liquid 
substances. 

122.  Capacities  in  Series  and  Parallel.  The  capacity  of  a 
condenser  in  the  form  of  two  parallel  plates  is  given  by  the 
formula 

(6)  C=M., 


where  A  is  the  area  of  one  plate, 
d  is  the  distance  between  the 
plates,  and  k  is  the  specific  in- 
ductive  capacity  of  the  dielectric.1 
Figure  80  represents  three  con- 

1  The  capacity  will  be  in  electrostatic  units  if  A  and  d  are  in  centimeters. 
If  the  result  is  to  be  expressed  in  microfarads,  the  factor  9  x  105  will  be  in- 
troduced into  the  denominator  of  equation  (6). 


170 


CAPACITY  AND  THE  CONDENSER     [V,  §  122 


densers  connected  in  parallel.  Since  the  capacity  of  a  con- 
denser is  proportional  to  the  area  of  its  plates,  it  follows  that 
the  capacity  (7,  equivalent  to  that  of  the  three  condensers,  is 
given  by 


c. 


Three  condensers  connected 
in  series  are  shown  in  Fig.  81. 
In  this  case  the  quantity  in 
each  condenser  is  the  same,  and 
is  equal  to  the  charge  which 
enters  the  system  from  the 
battery  B.  Moreover,  if  a  po- 
tential difference  V  is  applied  at  the  terminals  AB,  we  have 

where  vl}  vz,  and  v3  are  the  potential  differences  between  the 
plates  of  the  three  condensers,  respectively.     Hence,  we  have 


FIG.  81. 


(9) 
v  / 


- 
r< 

v> 


Since  Qi  =  $2  =  Qs>  the   equation  (9)  may  be  written  in  the 
form 


(10) 


V=Q 


\_GI    c/2    c/sj 


But  V=Q/C9  where  C  is  the  equivalent  capacity  and  Q  is  the 
charge  in  one  condenser  ;  hence, 


(11) 
and 
(12) 


c 


1-1+1+ 
c    c,    o, 


V,  §  123]    CHARGE,   CURRENT,   AND  ENERGY  171 

PART  II.     CHARGE,  CURRENT,  AND  ENERGY 
RELATIONS 

123.  The  Variation  of  Charge  with  Time.  Assume  a  con- 
stant potential  difference  V  impressed  on  a  circuit  which  con- 
tains a  capacity  C  and  non-inductive  resistance  R  (Fig.  82). 
It  is  to  be  understood  that  the  resistance  R  includes  all  the 
ohmic  resistance  throughout  the  entire  cir- 
cuit. The  condenser  does  not  instantly  ac- 
quire its  full  charge  on  closing  the  key  K, 
nor  is  the  discharge  an  instantaneous  process.  X  A 
It  is  important  to  investigate  the  time  rela-  C 
tions  of  charge  and  current  during  the  process  1 

T^IYl      R9 

of  charging  and  discharging  the  condenser. 

The  available  potential  difference  V,  which  is  assumed  to  be 
constant,  will  be  divided  into  two  parts.  One  part  V\  will 
maintain  the  current  strength  through  the  ohmic  resistance  R, 
and  the  other  part  F2  will  appear  at  the  condenser  terminals 
and  store  energy  in  the  form  of  charge.  Neither  of  these 
values  is  constant,  but  the  sum  of  the  two  is  constant,  and 
always  equal  to  V.  We  may  then  write 

(13)  F=F!+F2. 

Since  V\  and  F2  are  both  varying  continually,  their  instanta- 
neous values  must  be  used.  The  potential  difference  which 
maintains  current  through  R  is  always  iR,  and  the  instanta- 
neous value  of  i  is  dQ/dt ;  whence  we  have 

(14)  F'  =  -R? 

The  potential  difference  at  the  condenser  terminals  at  any 
instant  is  given  by  the  formula 

(15)  ^  =     i 


172  CAPACITY  AND   THE   CONDENSER     [V,  §  123 

whence,  throughout  the  period  of  charge  or  discharge, 

(16)  "-*£+§. 

Assuming  that  the  condenser  is  being  charged,  equation  (16) 
may  be  integrated,  and  expressions  may  be  found  for  the  values 
of  charge  and  current  at  any  time  t  seconds  after  closing  the 
key  K.  Separating  the  variables,  equation  (16)  becomes 


Integrating  this  expression  between  the  limits  zero  and  Q, 
and  remembering  that  Q  =  0  when  t  =  0,  we  have 


-  7?r  C    d® 
ICJ  VC^- 


where  e  is  the  base  of  the  Napierian  system  of  logarithms. 
Solving  this  equation  for  Q,  we  have, 

(19)  Q=VC-VCe-"*c. 

From  equation  (19)  it  is  seen  that  for  t  =  0,  Q  =  0,  which  was 
the  original  assumption  ;  but  if  £  =  oo  ,  then  Q  =  VC,  which 
represents  the  maximum  and  final  value  of  the  charge  in  the 
condenser. 

As  an  illustration  of  the  foregoing  relations,  consider  a  con- 
denser of  10  microfarads  capacity,  which  has  a  charging  poten- 
tial of  1000  volts  suddenly  applied  to  its  terminals,  the  circuit 
resistance  being  200  ohms.  The  final  value  of  the  charge  after 
an  infinite  time  is  given  by  equation  (19), 

Q  =  VC=  1000  X  10  x  10-6  =  0.01  coulomb. 

Choosing  intervals  of  time  of  0.001  second,  and  substituting 
these  values  for  t  in  equation  (19),  the  charge  corresponding 
to  each  instant  of  time  may  be  found.  From  these  values  the 
curve  7,  Fig.  83,  is  drawn.  This  curve  shows  the  relation 
between  coulombs  and  time.  It  is  evident  that  practically  the 


V,  §  125]    CHARGE,   CURRENT,   AND  ENERGY 


173 


full  value  of  the  charge  is  reached  in  a  few  thousandths  of  a 
second,  although  its  full  value  is  not  reached  until  a  much 
longer  time  has  elapsed.     The  final  value  of  the  charge  is 
seen  to  be  quite  inde- 
pendent of  the  value    s.o  o. 
of  R. 


II 


124.  The  Time 
Constant.  It  is  fre- 
quently necessary  to 
compare  the  behavior 

of    condensers    with     « 

£ 

regard  to  the  quick-    i, 

ness  with  which  they 
acquire  their  charges.  | 
It  is  obviously  im-  I: 
possible  to  use  for 
this  purpose  the  total 
time  involved  in  the 
process,  since  this  is 
theoretically  infinite. 
Custom  has,  how- 
ever, sanctioned  the  use  of  a  certain  time  interval  called 
the  time  constant  of  the  circuit.  Its  value  is  RC  seconds. 
The  corresponding  value  of  Q  is  readily  derived  from  equation 
(19).  If  t  is  made  equal  to  RC  seconds,  then 


o.wi    £ifwe  in  seconds 
FIG.  83. 


0.005 


(20) 


Q  =  VG--VC, 


and  it  is  clear  that  at  this  time  after  closing  the  key,  the 
charge  has  risen  to  a  point  which  falls  short  of  its  final  value 
by  1/e  times  that  final  value,  that  is,  about  0.37  times  that  final 
value. 

125.  The  Variation  of  the  Charging  Current  with  Time. 

The  instantaneous  value  of  the  current  at  any  time  t  can  be 


174  CAPACITY  AND  THE   CONDENSER     [V,  §  125 

found  by  differentiating  equation  (19)  with  respect  to  the  time. 
This  gives 


or 

(21)  i  =  Z.e-*i*c. 

R 

At  the  outset,  when  t  =  0,  it  is  seen  that 


and  the  current  begins  to  flow  as  if  there  were  no  capacity  in 
the  circuit.  The  condenser  begins  to  show  its  effect,  however, 
as  time  increases,  and  when  t  =  oo  ,  i  =  0.  If  t  is  made  equal 
to  RC  seconds,  equation  (21)  takes  the  form 

(22)  ,_1£ 

which  shows  that  the  current  has  fallen  to  1/e  times  its  initial 
value  when  t  =  EC.  If  values  of  C,  R,  and  F,  as  given  in  the 
numerical  illustration  of  §  123,  are  substituted  in  equation  (21), 
the  current  may  be  calculated  for  any  time  t.  Corresponding 
values  of  current  and  time  are  plotted  in  curve  II,  Fig.  83, 
from  which  it  appears  that  the  initial  current  is  large,  but 
that  it  rapidly  decreases  and  approaches  zero  as  the  charge 
approaches  its  final  value.  The  value  RC  seconds  which  was 
substituted  for  t  is  called  the  time  constant  of  the  circuit,  as 
stated  in  §  124. 

126.  The  Distribution  of  Energy.  In  order  to  study  the 
distribution  of  energy  in  a  circuit  during  the  process  of  charg- 
ing a  condenser,  we  shall  use  the  fact  that  energy  is  always 
given  by  the  product  of  the  potential  difference,  the  current, 
and  the  time.  The  energy  supplied  to  the  circuit  for  any 
short  interval  of  time  dt  is  therefore  given  by  the  equation 

(23)  dW=  Vidt, 


V,  §  126]    CHARGE,   CURRENT,   AND  ENERGY  175 

where  F  is  the  constant  impressed  potential  difference  and  i 
is  the  current  strength.  This  energy  may  be  set  equal  to  the 
sum  of  the  energy  dissipated  as  heat  in  the  ohmic  resistance, 
and  the  energy  stored  in  the  condenser.  Writing  this  equa- 
tion, we  have 

(24)  Vidt  =  VRdt  +  Q  i  dt. 

C 

If  the  instantaneous  values  of  i  and  Q  are  substituted  in 
the  two  terms  of  the  right-hand  member  of  (24),  these  terms 
may  be  separately  integrated'  between  limits  t  =  0  and  £  =  oo  , 
and  values  may  be  found  for  the  energy  transformed  into  heat, 
and  for  that  stored  in  the  condenser.  Substituting  the  value 
of  i  from  equation  (21),  the  term  i2JRdt}  which  we  shall  denote 
by  d  WB,  takes  the  form 

dWs  =  —   - 
R 

and 

(25)  WR 

From  this  equation  it  is  evident  that  the  total  energy  dissi- 
pated as  heat  in  the  ohmic  resistance  of  the  circuit  is  not  de- 
pendent upon  R,  but  only  upon  V  and  G. 

In  order  to  find  the  energy  stored  in  the  condenser,  we  may 
substitute  the  instantaneous  values  of  Q  and  i  from  equations 
(19)  and  (21),  in  the  last  term  of  (24),  and  integrate  between 
the  same  limits  as  before.  The  term  Qidt/C,  which  we  shall 
denote  by  dWGj  is,  therefore,  of  the  form 


whence,  we  find 

(26)          B^r=-~  f 
RJo 


dWc  =  -E!  [«- 
R 


176  CAPACITY  AND   THE   CONDENSER      [V,  §  126 

This  is  seen  to  be  exactly  the  same  result  as  that  of  equation 
(25).  Hence  it  appears  that  one  half  of  the  total  energy 
given  to  the  circuit  is  lost  as  heat  in  the  resistance,  and  one 
half  is  stored  in  the  condenser. 

The  charged  condenser  is  analogous  to  a  stressed  spring,  in 
which  energy  is  stored  while  under  stress ;  an  equivalent 
amount  of  energy  is  returned  when  the  constraint  is  released. 

127.  The  Variation  of  Charge  and  Current  during  Dis- 
charge. Expressions  similar  to  those  of  the  preceding  articles 
may  be  derived  for  the  instantaneous  values  of  charge  and 
current  during  the  discharge  of  a  condenser.  Returning  to 
equation  (16),  and  setting  F  =  0,  which  means  that  the  im- 
pressed voltage  is  cut  off  and  the  circuit  left  to  itself,  we  have 

(27)  0  =  ^  +  |. 
Separating  the  variables,  we  find 

(28)  «»  =  _*Cdg. 

Integrating  this  between  the  limits  Q0  and  Q,  which  represent 
respectively  the  initial  and  final  values  of  the  charge,  we  have 


F«o 

or 

(29) 

Equation  (29)  may  be  put  in  the  exponential  form, 

(30)  Q=  QQe~t/RC. 

If  the  initial  value  of  the  charge  is  given  by  the  formula 

equation  (30)  becomes 

(31)  Q=VCe-t'*°. 


V,  §  127]     CHARGE,   CURRENT,   AND  ENERGY  177 

This  gives  the  value  of  the  charge  in  the  condenser  at  any 
time  t  seconds  after  discharge  begins. 

To  find  the  value  of  the  current  at  any  instant  during  the 
discharge,  equation  (31)  is  differentiated  with  respect  to  t, 
which  gives 


This  is  seen  to  be  the  same  expression  as  that  for  the  charg- 
ing current  given  in  equation  (21),  except  that  the  sign  is 
negative,  which  indicates  a  reversed  direction. 

The  discussion  in  the  preceding  articles  shows  that  a  con- 
denser acquires  its  charge  according  to  an  exponential  function 
of  the  time.  The  charge  reaches  its  final  and  maximum  value 
theoretically  only  after  an  infinite  period,  although  in  most 
condensers  in  actual  use  the  charge  is  practically  complete  in 
a  fraction  of  a  second.  The  final  value  of  the  charge  is  quite 
independent  of  the  resistance  of  the  circuit.  Through  its 
influence  on  the  current,  however,  the  resistance  does  control 
the  rate  at  which  the  charge  is  stored  or  given  up.  Moreover, 
neither  the  energy  lost  as  heat  in  the  resistance,  nor  that 
stored  in  the  condenser,  depends  on  the  actual  value  of  the 
resistance.  It  may  be  shown  also  that  the  rates  of  storing 
and  giving  up  energy  are  quite  different  in  the  processes  of 
charging  and  discharging. 

EXERCISES 

1.  A  condenser  has  a  capacity  of  0.3  mf.,  and  is  charged  with  a 
potential  difference  of  1.434  volts.     Calculate  the  value  of  its  charge  in 
(a)  coulombs,  (6)  microcoulombs,  (c)  C.  G.  S.  units. 

2.  Derive  the  dimensional  formula  for  capacity. 

3.  Check  the  equations  of  §§  123-127  by  means  of  dimensional  formulas. 


CHAPTER  VI 
ELECTROMAGNETIC   INDUCTION 

PART  I.     MUTUAL  AND  SELF  INDUCTANCE 

128.  The  Linking  of  Circuits  with  Lines  of  Force.    A 

straight  wire  carrying  a  current  is  known  to  have  about  itself 
a  magnetic  field.  The  general  form  of  the  lines  of  force  is  a 
circle,  concentric  about  the  axis  of  the  wire.  The  existence 
of  this  field  is  an  indication  of  current  flowing  along  the  wire, 
and  the  intensity  of  the  field  has  been  shown  to  bear  a  direct 
ratio  to  the  current  strength.  (See  equation  (8),  Chapter  IV.) 
The  line  of  force  must  be  regarded  as  a  closed  curve  along 
which  a  magnetic  pole  will  move.  From  whatever  point  the 
pole  may  start,  it  will  return  again  to  the  same  point.  If  the 

line  of  force  is  due  to  a 
bar  magnet,  or  to  an  elec- 
tromagnet, part  of  the  path 
will  be  through  iron  and 
part  through  air. 

FIG  84        ^~r  Moreover,    the    electric 

circuit  is   itself   a   closed 

curve  along  which  the  electric  charge  passes  from  the  generator 
back  to  the  generator  again.  The  lines  of  force  must  then  be 
considered  as  linked  with  the  electric  circuit,  as  represented 
in  Fig.  84. 

In  considering  the  linking  of  lines  of  force  with  an  electric 
circuit,  two  different  cases  may  be  distinguished :  (1)  when 
the  circuit  is  originally  without  current,  and  is  brought  into  a 

178 


VI,  §  128]     MUTUAL  AND  SELF  INDUCTANCE  179 

magnetic  field;  (2)  when  the  circuit  conveys  the  current  which 
produces  the  magnetic  field. 

In  the  first  case,  suppose  a  closed  loop  of  a  single  turn  of 
wire  not  carrying  a  current  is  brought  into  the  neighborhood 
of  a  bar  magnet.  Some  of  the  lines  of  force  of  the  magnetic 
field  will  link  with  the  loop.  If  <£  lines  of  force  thus  link  with 
the  single  turn,  the  number  of  linkings  is  given  by  N  =  <f>.  If 
there  are  S  turns  of  wire  linking  with  <f>  lines  of  force,  then 
the  total  number  N  of  such  linkings  is  given  by  the  equation 

(1)  N=S<t>. 

In  the  second  case,  let  <f>  represent  the  total  number  of  lines 
of  force  due  to  the  flowing  current  which  thread  through  the. 
circuit  (Fig.  84).  If  there  is  a  single  turn  of  wire,  the  number 
of  linkings  is  given  by  N  =  <£.  If,  however,  there  are  S  wire 
turns,  each  line  of  force  is  considered  as  linking  with  every 
turn.  The  total  number  N  of  such  linkings  is  given  by  the 
product  of  the  number  of  wire  turns  and  the  number  of  mag- 
netic lines ;  that  is, 

(2)  N=S<j>. 

This  product  of  magnetic  flux  lines  by  wire  turns  is  frequently 
called  flux  turns. 

It  is  important  to  consider  in  what  ways  the  number  of 
linkings,  or  flux  turns,  may  be  changed.  Assuming  that  the 
permeability  of  the  medium  is  constant,  suppose  first  that  a 
loop  of  a  conductor  not  carrying  current  is  placed  in  a  mag- 
netic field.  The  number  of  linkings  may  be  changed  by 
changing  (a)  the  field  strength,  (6)  the  position  of  the  loop, 
(c)  the  dimensions  or  shape  of  the  loop,  (d)  the  number  of 
turns  of  wire. 

In  the  case  of  a  closed  circuit  which  does  carry  current  the 
number  of  linkings  may  be  changed  by  changing  (a)  the  cur- 
rent strength,  (6)  the  number  of  turns  of  wire,  (c)  the  shape 
or  dimensions  of  the  circuit. 


180 


ELECTROMAGNETIC  INDUCTION     [VI,  §  129 


Faraday  first  showed  that  any  change  in  the  number  of 
linkings  between  wire  turns  and  flux  lines  gives  rise  to  an 
induced  electromotive  force.  Moreover,  Lenz's  law  states 
that  the  induced  current  arising  from  this  E.  M.  F.  is  always 
so  directed  as  to  oppose  the  change.  Lenz's  law  is  merely  the 
statement  of  the  principle  of  the  conservation  of  energy  for 
the  electrical  case.  From  this  principle  it  is  apparent  that 
the  energy  put  into  the  circuit  to  bring  about  a  change  in  the 
number  of  linkings  is  precisely  equal  to  the  energy  of  the 
induced  current  arising  from  the  change. 

129.  The  Faraday  Equation.  Assume  a  conductor  bent 
into  the  form  ABC,  Fig.  85,  and  lying  in  the  magnetic  field 

N8,  with  its  plane 
perpendicular  to  the 
lines  of  force.  On 
the  horizontal  and 
parallel  wires  A  and 
C  lies  a  bar  a  b  which 
can  move  in  either 
direction  along  the 
wires,  and  is  always 
in  contact  with  them. 
Suppose  the  bar  is 
moved  to  the  right 
through  a  short  dis- 
tance dx  in  a  time 
interval  dt,  thus  cut- 


ting across  the  lines 
of  force,  and  changing 

the  number  of  linkings  between  the  circuit  and  the  field. 
This  motion  sets  up  an  induced  current  in  «6,  in  the  direction 
from  b  to  a.  If  V  is  the  induced  electromotive  force, 
and  i  the  instantaneous  value  of  the  induced  current,  then 


FIG.  85. 


VI,  §  129]     MUTUAL  AND  SELF  INDUCTANCE  181 


the  energy  dW^  of  the  induced  current  is  given  by  the 
equation 

(3)  dWl=Vidt. 

Let  H  be  the  value  of  the  magnetic  field  strength  and  I  the 
length  of  ab  between  A  and  C.  Then,  when  a  current  i  flows 
in  ab,  there  will  be  a  force  acting  on  it  given  by  the  equation 

(4)  F=ilH. 

When  the  bar  is  moved  through  a  distance  dx  against  this 
force,  the  work  dW2  done  is  given  by  the  equation 

(5)  dW2  =  ilHdss. 

Since  from  Lenz's  law  the  work  done  in  moving  the  conductor 
is  equivalent  to  the  energy  associated  with  the  current  in- 
duced, the  expressions  (3)  and  (5)  may  be  equated,  and  we 
have 

(6)  Vidt  =  -ilHdx. 

The  negative  sign  shows  that  the  current  induced  is  directed 
so  that  its  magnetic  field  reacts  with  the  field  H  to  oppose  the 
motion,  as  required  by  the  law  of  conservation  of  energy.  In 
place  of  Idx,  which  is  an  area,  dA  may  be  written,  and  the 
product  HdA  gives  the  change  in  the  number  of  linkings  dN, 
due  to  the  motion  of  ab.  Equation  (6)  may  then  be  written 

in  the  form 

Vi  dt  =  -i  dN, 
or 

(7)  F=-f. 

This  expression  is  called  Faraday's  equation.  It  states  that 
the  induced  potential  difference  is  numerically  equal  to  the 
time  rate  of  change  of  the  number  of  linkings.  One  C.  G.  S. 
unit  is  induced  when  one  line  of  force  is  cut  by  one  wire  in 
one  second.  In  order  to  induce  an  electromotive  force  of  one 
volt  in  the  circuit,  108  lines  of  force  must  be  cut  per  second 
by  a  single  wire. 


182  ELECTROMAGNETIC   INDUCTION     [VI,  §  130 

130.  Mutual  Inductance  of  Two  Circuits.  Consider  a  cir- 
cuit A  (Fig.  86)  carrying  a  current  of  strength  i.  The  mag- 
netic field  strength  at  any  point  in  the  neighborhood  of  A  is 

proportional  to  i  (§  104).  Since 
the  field  strength  at  any  point  is 
measured  by  the  number  of  lines 
of  force  per  unit  area,  it  follows 
that  the  number  of  lines  through 
any  chosen  area  near  A  is  propor- 
tional to  i. 

Suppose  that  another  conductor  forming  a  closed  circuit  B 
is  near  A,  and  that  the  number  of  linkings  between  wire  turns 
of  circuit  B  and  lines  of  force  due  to  circuit  A  is  equal  to  N. 
The  number  N  is  proportional  to  i,  and  if  the  relative  posi- 
tions of  A  and  B,  the  number  of  wire  turns  in  each,  and  the 
permeability  of  the  surrounding  medium  are  not  changed, 
then 

(8)  N=  Mi, 
or 

(9)  Jf=^, 

i> 

where  the  factor  M  is  a  geometric  constant  of  the  pair  of  cir- 
cuits, which  is  quite  independent  of  the  current  strength. 

If  the  current  in  the  circuit  A  changes  by  an  amount  di,  the 
number  of  linkings  with  the  B  circuit  will  change  by  a  corre- 
sponding amount  dN,  whence 

(10)  jr-f. 

Moreover,  if  this   change   takes  place  in  a  time  dt,  we  may 
write 


dt 


VI,  §  131]     MUTUAL  AND  SELF  INDUCTANCE  183 

It  has  been  shown  in  equation  (7)  that  the  time  rate  of  change 
of  linkings  gives  the  value  of  the  induced  potential  difference. 
Whence,  the  equation  (11)  may  be  written  in  the  form 


at 

This  constant  M  is  called  the  mutual  inductance  of  the  two 
circuits,  and  it  may  be  defined  in  any  one  of  the  following 
ways  :  (a)  from  equation  (9),  M  is  numerically  equal  to  the  total 
number  of  linkings  with  the  B  circuit,  when  unit  current  flows  in 
the  circuit  A;  (b)  from  equation  (10),  M  is  numerically  equal 
to  the  change  in  the  number  of  linkings  with  the  B  circuit,  when 
the  current  in  the  A  circuit  is  changed  by  unit  amount;  (c)  from 
equation  (11),  M  is  the  factor  determined  by  the  constant  ratio 
between  the  time  rate  of  change  of  Unkings,  and  the  time  rate  of 
change  of  current  strength;  (d)  from  equation  (12),  M  is  the 
value  of  the  potential  difference,  or  E.  M.  F.,  induced  in  the  B 
circuit,  when  the  current  in  the  A  circuit  is  changing  at  unit  rate. 

131.  Units  and  Standards  of  Mutual  Inductance.    The 

practical  unit  of  mutual  inductance  is  called  the  henry :  a  pair 
of  circuits  has  a  mutual  inductance  of  one  henry  when  an  E.  M.  F. 
of  one  volt  is  induced  in  one  of  them  if  a  change  of  one  ampere 
per  second  occurs  in  the  other.  The  henry  is  equal  to  109  C.  Gr.  S. 
units. 

The  mutual  inductance  of  two  circuits  can  be  computed 
only  when  the  number  of  lines  of  force  arising  in  one  circuit, 
and  linking  with  the  other  circuit,  is  known.  This  is  possible 
only  in  a  few  simple  cases. 

One  arrangement  for  which  the  mutual  inductance  may  easily 
be  calculated  is  shown  in  cross-section  in  Fig.  87.  A  long  bar 
of  wood  or  hard  rubber  AA'  is  turned  to  a  uniform  diameter, 
except  for  a  short  distance  at  the  middle  aa',  where  the  diameter 


184  ELECTROMAGNETIC  INDUCTION     [VI,  §  131 

is  made  slightly  less  than  that  of  the  rest  of  the  bar.  In  this 
middle  channel  is  wound  a  large  number  of  turns,  S,  of  fine 
wire,  usually  a  thousand  or  more,  the  ends  being  brought  out 

. QQQQQOOOOOOOOOQOOOO 

OOOOOOOOO~~ 7 


OOOOOOOOOOOOOOOOOOO 
Fia.  87. 

to  terminal  binding  posts.  Outside  of  this  coil,  and  running 
from  end  to  end  of  the  core,  is  wound  in  a  uniform  layer  the 
coil  P. 

For  the  long  coil,  called  the  primary,  T  represents  the  total 
number  of  turns,  L  is  the  length,  and  T/L,  or  n,  is  the  number 
of  turns  per  centimeter  of  length.  For  the  short  secondary 
coil,  S  is  the  total  number  of  turns,  r  is  the  radius  of  one  turn, 
and  A  is  the  area  of  cross-section. 

Remembering  that  the  mutual  inductance  of  two  circuits  is 
given  by  the  number  of  linkings  established  when  unit  current 
flows,  the  value  of  M  for  the  present  case  is  easily  calculated. 
Let  a  current  of  strength  i  C.  G.  S.  units  flow  through  the 
primary  coil.  The  value  of  the  magnetic  field  strength  at  the 
center  of  the  coil  is,  from  equation  (33),  Chapter  IV, 

(13)  H=±Trni. 

Since  this  may  be  regarded  as  uniform  over  the  entire  cross- 
section,  the  total  magnetic  flux  through  the  secondary  coil  is 
given  by  the  formula 

(14)  <j>  =  HA  =  4  imiA  =  4  ifriir*. 

All  of  these  lines  interlink  with  all  the  turns  of  the  secondary ; 
hence,  by  equation  (1),  the  total  number  of  linkings  is 

(15)  N=4>S  = 


VI,  §  131]     MUTUAL  AND  SELF  INDUCTANCE  185 

From  equation  (9)  we  have 

i  ' 

substituting  in  this  the  value  of  N  from  equation  (15),  we  have 

(16)  M  =  4  Tr^nr^S  C.  G.  S.  units, 
or 

(17)  M  =    vn* —  henrys. 

This  form  of  mutual  inductance,  which  is  so  carefully  con- 
structed that  its  value  can  be  calculated  from  its  dimensions, 
is  called  a  standard  current  inductor.  The  ratio  of  the  length 
to  the  radius  should  not  be  less  than  fifty.  The  secondary 
coil  is  sometimes  wound  outside  of  the  primary,  in  which  case 
the  area  of  the  cross-section,  by  which  the  value  of  the  flux 
density  is  multiplied  in  (14),  is  that  of  the  primary  coil.  The 
secondary  coil  should  lie  close  to  the  primary  to  avoid  leakage. 

The  equation  (16)  was  obtained  on  the  assumption  that  the 
permeability  of  the  medium  is  unity.  If  any  magnetic  sub- 
stance of  permeability  p  is  introduced,  the  value  of  the  mutual 
inductance  becomes 

(18)  M  =  4  v*ni*Sii  C.  G.  S.  units. 

Since  the  permeability  changes  with  the  magnetic  field 
strength,  and  hence  with  the  current,  M  can  be  calculated  in 
this  way  only  when  the  corresponding  values  of  /*  and  H  are 
known.  For  this  reason,  it  is  customary  to  avoid  all  magnetic 
materials  in  the  construction  of  such  coils. 

A  more  convenient  laboratory  standard  with  any  desired 
value  is  made  by  winding  the  necessary  number  of  turns  on 
a  marble  spool.  The  winding  is  then  thoroughly  impregnated 
with  an  insulating  varnish  and  baked  hard.  After  mounting 
on  an  ebonite  or  wooden  base  it  is  carefully  calibrated. 
Standards  prepared  in  this  way  are  very  constant.  They 
should  be  occasionally  checked  in  a  standardizing  laboratory. 


186  ELECTROMAGNETIC   INDUCTION     [VI,  §  132 

132.  The  Mutual  Inductance  of  Symmetrical  Circuits.    If 

two  circuits  A  and  B  have  a  mutual  inductance  M,  it  can  be 
shown  that  the  number  of  linkings  with  B  due  to  unit  current 
in  A  is  exactly  equal  to  the  number  of  linkings  with  A  due  to 
unit  current  in  B.  The  general  proof  of  this 
Y  requires  more  powerful  methods  of  analysis  than 

those  introduced  thus  far,  but  in  the  case  of  two 
!•** 
I      symmetrical  circuits,  the  proof  is  simple  and  is  as 

^          fo     follows. 
FIG.  88.  Consider  two  circuits  A  and  B,  shown  in  cross- 

section  in  Fig.  88,  which  are  in  every  way  sym- 
metrical. Let  MI  be  the  mutual  inductance  when  a  current 
flows  through  A,  and  let  M2  be  their  mutual  inductance  when 
the  same  current  flows  through  B.  For  a  current  of  strength  i 
in  A,  the  number  of  linkings  with  B  is  given  by  the  equation 

(19)  Ni  =  Mj. 

For  the  same  current  through  B,  the  number  of  linkings  with 
A  is  given  by  the  equation 

(20)  N2  =  M2i. 
By  symmetry  we  have 

whence 

(21)  Ml  =  M2. 

133.  Self-inductance  of  a  Circuit.     With  a  single  circuit 
carrying  a  current  i,  as  shown  in  Fig.  84,  the  number  of  lines 
of  force  threading  through  the  circuit,  and  hence,  the  number 
N  of  linkings,  is  proportional  directly  to  the  current  strength. 
This  fact  may  be  expressed  by  the  equation 

(22)  N  =  Li, 
or 

(23)  L  =  ~- 


VI,  §  134]     MUTUAL  A-ND  SELF  INDUCTANCE  187 

If  the  current  changes  by  some  small  amount  di,  the  number 
of  linkings  will  change  by  some  corresponding  amount  dN, 
whence 

(24)  L=«j. 

Moreover,  if  this  change  takes  place  in  a  time  dt,  we  may  write 

(25) 


_dt      dt 

Since  by  equation  (7)  the  time  rate  of  change  of  the  number  of 
linkings  gives  the  induced  potential  difference,  the  equation 
(25)  may  be  written  in  the  form 


dt 

If  the  geometric  form,  the  number  of  wire  turns,  and  the 
permeability  of  the  medium  are  constant,  L  is  itself  constant, 
and  quite  independent  of  the  current  strength.  This  quantity 
L  is  called  the  self  -inductance  of  the  circuit,  and  it  is  de- 
nned in  any  one  of  the  following  ways  :  (a)  from  equation 
(23),  L  is  numerically  equal  to  the  total  number  of  Unkings,  when 
unit  current  is  flowing  ;  (b)  from  equation  (24),  L  is  numerically 
equal  to  the  change  in  the  number  of  linkings  when  the  current  is 
changed  by  unit  amount  ;  (c)  from  equation  (25),  L  is  the  factor 
determined  by  the  constant  ratio  between  the  time  rate  of  change 
of  linkings  and  the  time  rate  of  change  of  current;  (d)  from 
equation  (26)  L  is  the  value  of  the  potential  difference,  or  E.  M.  F. 
induced  in  the  circuit  when  the  current  changes  at  unit  rate. 

134.  Units  and  Standards  of  Self-inductance.  The  prac- 
tical unit  of  self-inductance  is  the  henry  :  a  circuit  has  an  in- 
ductance of  one  henry  when  one  volt  is  induced  at  its  terminals  by 
a  change  in  current  strength  of  one  ampere  per  second.  The 
millihenry  and  microhenry  are  convenient  subdivisions  of  the 
henry.  (See  §  12.) 


188  ELECTROMAGNETIC  INDUCTION     [VI,  §  134 

Taking  any  one  of  the  denning  equations  for  self-inductance, 
its  dimensional  formula  is  found  to  be 


(27)  L 

hence  its  absolute  unit  should  be  the  same  as  the  unit  of 
length,  the  centimeter.  The  henry  is  equivalent  to  109 
centimeters.1 

The  secondary  circuit  of  a  f-inch  induction  coil  will  have  an 
inductance  of  approximately  15  henry  s  ;  an  ordinary  telegraph 
sounder  20  to  30  millihenry  s  ;  and  the  coils  of  a  sensitive,  sus- 
pended needle  galvanometer  from  one  to  two  henrys.  Good 
resistance-box  coils  should  have  an  inductance  of  less  than  one 
microhenry.  The  inductance  of  a  dynamo  field  magnet  may 
exceed  1000  henrys.  The  primary  of  an  induction  coil  20 
inches  long  will  have  an  inductance  of  approximately  20 
henrys,  while  the  secondary  of  such  a  coil,  with  a  resistance  of 
30,000  ohms,  may  have  an  inductance  of  2000  henrys. 

As  in  the  case  of  mutual  inductance,  the  calculation  of  the 
self-inductance  of  circuits  from  their  dimensions  is  feasible 
only  in  a  few  cases.2  One  form  of  circuit  for  which  this  calcu- 
lation is  easily  made  is  the  long  solenoid.  Assume  a  long 
solenoid  of  radius  r,  area  of  cross-section  A,  and  length  /,  with 
T  total  turns  of  wire,  and  n  turns  per  centimeter.  From 
equation  (33),  §  109,  the  value  of  the  field  strength  at  the 
center  of  the  coil  is  given  by 

(28)  JT=4imi, 

and  the  total  flux  across  the  central  cross-sectional  plane  is 

(29)  4  =  HA  =  4  imiA  =  4  7rznirz. 

1  Since  the  distance  along  the  arc  of  a  great  circle  of  the  earth,  from  the 
equator  to  the  north  pole,  was  originally  taken  as  109  centimeters,  an  early 
name  for  the  practical  unit  oi  inductance  was  the  quadrant. 

2  The  formulas  given  here  for  calculating  self  and  mutual  inductances  from 
the  dimensions  of  the  coils  are  only  approximate.     Accurate  formulas,  which 
are  rather  complicated,  are  given  in   the   BULLETIN   OF  THE  BUREAU  OF 
STANDARDS,  Vol.  8,  1912.     Approximate  formulas  for  coils  of  various  shapes 
will  be  found  in  the  various  electrical  handbooks. 


VI,  §  134]     MUTUAL  AND  SELF  INDUCTANCE  189 

Assuming  that  all  of  these  lines  of  force  link  with  all  of  the 
wire  turns,  which  is  very  nearly  true,  the  total  number  of 
linkings  is  given  by  the  formula 

(30)  N=<j>T=±'n*nirzT. 
Since  T  =  nl,  equation  (30)  becomes 

(31)  N=  4  TrWiM. 
From  equation  (23),  we  have 

(32)  L  =  ^=  4  TrWrH  C.  G.  S.  units, 
or 

(33)  £ 


If  the  permeability  of  the  medium  within  the  coil  is  /*,,  equa- 
tion (32)  becomes 

(34)  Z  =  47r2n2rV 

However,  in  the  construction  of  self-inductance  coils  which  are 
to  be  used  as  standards,  magnetic  materials  are  avoided,  be- 
cause of  .the  variation  of  the  magnetic  permeability  with  cur- 
rent strength. 


190  ELECTROMAGNETIC   INDUCTION     [VI,  §  135 

PART  II.     CURRENT,  ENERGY,  AND  CHARGE 
RELATIONS 

135.   Current  and  Energy  Relations  in  Inductive  Circuits. 

A  piece  of  matter  cannot  set  itself  in  motion,  but  requires 
energy  from  without  to  effect  a  change  in  its  momentum. 
Similarly  an  electric  current  cannot  set  itself  in  motion,  and 
energy  must  enter  the  circuit  in  order  that  electric  charge 
may  be  transferred.  It  has  been  shown  in  §  129  how  energy 
can  enter  a  circuit  by  making  lines  of  force  cut  that  circuit,  so 
as  to  change  the  number  of  linkings  with  the  wire  turns. 

When  a  current  is  established,  a  magnetic  field  is  created  in 
the  ether,  the  medium  about  the  circuit,  and  when  the  current 
ceases,  this  magnetic  field  collapses  and  disappears.  This 
magnetic  field  is  of  the  nature  of  a  strained  condition  in  the 
medium,  and  work  is  required  to  establish  the  field.  More- 
over, when  such  a  field  collapses  or  disappears,  the  stored 
energy  is  returned  to  the  circuit  in  the  form  of  an  induced 
current. 

The  magnetic  field  about  a  circuit  carrying  a  current  is  then 
to  be  considered  as  the  seat  of  a  certain  amount  of  energy 
whose  magnitude  depends  upon  the  strength  of  the  current 
and  also  upon  its  distribution,  that  is,  upon  the  shape  and 
dimensions  of  the  circuit  and  the  number  of  turns  of  wire. 
This  is  quite  analogous  to  the  case  of  a  system  of  material 
particles  in  rotation,  in  which  the  kinetic 
energy  of  rotation  depends  upon  the  masses 
of  the  particles  as  well  as  upon  their  space 
distribution. 

Consider  a  circuit  (Fig.  89)  in  which  the 
inductance  and  resistance  are  concentrated  in 
L   and    R    respectively,   with   an   impressed 
potential  difference  V.     In  a  non-inductive  circuit  the  current 
rises  to  its  full  value  instantly  when  the  circuit  is  closed. 


V  ->- 


VI,  §  136]     CURRENT,   ENERGY,   AND  CHARGE          191 

This  is  not  the  case,  however,  when  an  inductance  is  present, 
for  in  this  case  magnetic  flux  lines  are  linked  with  the  wire 
turns  of  the  circuit.  We  have  seen  that  this  means  that  a 
potential  difference  is  established  which  depends  on  the  rate 
of  change  of  current,  and  which  opposes  the  change.  When 
the  key  K  is  closed  and  current  begins  to  flow  through  L>  its 
associated  field  sweeps  out  and  cuts  other  wire  turns,  thus 
setting  up  a  potential  difference  given  by  equation  (26), 

(35)  V  =  ~L- 


136.  The  Helmholtz  Equation.  The  impressed  potential 
difference  V  (Fig.  89)  may  be  considered  as  at  any  instant 
equal  to  the  sum  of  two  components  :  one  part  V\  maintains 
the  current  in  the  ohmic  resistance  of  the  circuit,  while  the 
other  part  F2  maintains  the  growing  current  in  the  inductance. 
We  may  then  write 

F=  7i  +  F2 
or 

(36)  V=iR  +  L-- 

Equation  (36)  is  called  the  Helmholtz  equation.  It  gives  the 
instantaneous  value  of  the  potential  difference  at  the  terminals 
of  an  inductive  circuit.  The  inductance  L  is  always  essentially 
a  positive  quantity  and  may  be  equal  to  or  greater  than  zero. 

Suppose  L  =  0  ;  then  the  second  term  of  the  right-hand 
member  of  equation  (36)  is  zero,  and  the  impressed  potential 
difference  is  equal  to  the  iR  drop  through  the  circuit. 

Suppose  that  L  is  greater  than  zero  ;  then  for  a  rising  value 
of  i,  that  is,  with  di/dt  positive,  it  is  obvious  that  the  iR  drop 
through  the  circuit  is  not  so  large  as  the  impressed  F",  and 
the  quantity  L  (di/dt)  is  of  the  nature  of  a  reversed  potential 
difference  which  retards  the  rise  in  the  current. 


192  ELECTROMAGNETIC   INDUCTION     [VI,  §  136 

For  a  decreasing  current  the  term  L(di/dt)  is  negative; 
then  the  iR  drop  is  greater  than  the  impressed  V.  This 
means  that  the  instantaneous  value  of  the  current  is  greater 
than  it  would  be  in  a  non-inductive  circuit,  and  the  effect  of  L 
is  to  retard  the  fall  in  current  strength. 

If  the  current  strength  is  constant,  that  is,  if  di/dt  =  0,  the 
impressed  V  is  equal  to  the  iR  drop. 

It  is  then  obvious  that  in  a  circuit  possessing  inductance, 
through  which  a  varying  current  is  flowing,  the  impressed 
voltage  is  not  represented  by  the  iR  drop  in  the  circuit,  but 
includes  also  that  part  which  is  utilized  in  storing  energy  in 
the  magnetic  field  about  the  circuit.  The  effect  of  the  induc- 
tance is  to  make  the  changes  in  the  current  lag  behind  the 
changes  in  the  impressed  potential  difference. 

Self-inductance  is  thus  seen  to  be  that  property  of  the  circuit 
which  opposes  any  change  that  is  made  in  the  current  strength. 
It  is  analogous  to  the  inertia  of  matter,  which  opposes  any 
changes  in  the  velocity  impressed  on  a  body. 

The  Helmholtz  equation  may  be  written  in  the  form 


This  equation  shows  that  during  the  period  of  increase  of  i, 
the  instantaneous  value  of  the  current  is  less  than  the  final 
value  which  V  is  able  to  maintain  by  the  amount 

LM. 

Rdt 

The  theory  of  dimensions  requires  that  L  (di/dt)  must  itself 
be  a  potential  difference,  and  the  negative  sign  shows  that  it 
tends  to  retard  the  growth  of  i.  If  di/dt  is  negative,  as  it  is 
for  a  decreasing  current,  the  sign  of  the  last  term  of  equation 
(37)  is  positive;  hence,  i  is  greater  than  its  steady  initial 
value  V/R. 

In  large  distributing  systems  which  carry  heavy  currents, 


VI,  §  137]     CURRENT,   ENERGY,   AND  CHARGE          193 

and  in  which  the  time  constants  are  small,  the  effects  of  sudden 
breaks  or  short  circuits  are  often  exceedingly  violent.  Enor- 
mous induction  currents  are  quickly  established,  which  have  the 
suddenness  and  disastrous  effects  of  violent  explosions. 

137.  The  Growing  Current  and  the  Time  Constant.     Let 

us  assume  that  a  potential  difference  V  is  suddenly  impressed 
on  a  circuit  of  resistance  R  and  inductance  L.  The  Helmholtz 
equation, 

(38)  V=iR  +  L^f, 

is  a  differential  equation  which  expresses  the  general  relation 
between  the  variable  quantities  i  and  t.  The  integration  of 
this  equation  gives  the  instantaneous  value  of  the  current  at 
any  time  t  seconds  after  the  potential  difference  is  impressed. 
Separating  the  variables,  we  have 

(39)  SX-V i|, 

«     HI  -!=-!!      :     | 

whence 

di  R 

(41) 


The  final  value  of  the  current  after  it.  has  become  steady  is 
V/R ;  let  us  denote  this  by  /.  Integrating  between  the  limits 
of  zero  and  t,  at  which  instants  the  currents  are  zero  and  i, 
respectively,  we  obtain 

/**      7? 
(42)  /    ,      F=Jo  —i*> 


/          F 
•*         J 


whence 
(43) 


194 


ELECTROMAGNETIC   INDUCTION     [VI,  §  137 


or 

(44)  loge(i  —  /) —  loge( —  7) 

±j 
or 

(45)  loge = 1, 

or 

(46)  i=I(l  —  e-*t/L). 

From  equation  (46)  it  is  seen  that  when  t  =  0,  i  =  0,  and  when 
t  equals  infinity,  i  =  I.  The  general  form  of  the  curve  that 
shows  the  relation  between  values  of  i  and  t  is  shown  in  Fig. 
90,  curve  I,  which  is  asymptotic  to  the  line  i  =  /.  Only  after 


FIG.  90. 

a  time  which  is  theoretically  infinite,  does  i  become  strictly 
equal  to  /;  but  for  values  of  L  and  R  which  are  actually  in 
use,  i  practically  attains  its  final  value  in  a  few  seconds,  or 
even  in  a  fraction  of  a  second. 

In  order  to  compare  the  quickness  with  which  currents  grow 
in  inductive  circuits,  it  is  obviously  impossible  to  use  the  full 
time  required  for  attaining  the  steady  state,  as  these  times 


VI,  §  138]     CURRENT,   ENERGY,  AND  CHARGE          195 

are  in  every  case  theoretically  infinite.  It  is  customary,  how- 
ever, to  use  the  time  required  for  the  current  to  reach  some 
definite  fraction  of  its  final  value.  If  t  is  made  equal  to  L/R 
seconds  in  equation  (46),  that  equation  takes  the  form 

(47)  i=I-^I> 

and  it  is  seen  that  at  this  time  the  current  falls  short  of  its 
full  or  final  value  by  an  amount  equal  to  1/e  times  the  final 
value,  that  is,  about  0.37  times  the  final  value.  This  factor 
L/R  is  a  characteristic  constant  of  the  inductive  circuit,  and  is 
called  the  time  constant.  The  effect  of  the  inductance  in 
delaying  the  rise  of  the  current  to  its  full  value  is  measured 
in  terms  of  this  time  constant. 

138.  The  Falling  Current  and  the  Time  Constant.  Con- 
sider a  circuit  of  resistance  R  and  inductance  L,  in  which  a 
steady  current  IQ  is  being  maintained  by  an  electromotive  force 
of  value  V.  If  the  electromotive  force  is  suddenly  cut  off  by 
the  opening  of  a  switch,  the  relation  between  the  decreasing 
current  and  the  time  is  given  by  the  integration  of  the  Helrn- 
holtz  equation,  which  for  this  case  will  be  written  in  the  form 

(48)  0  =  Ri  +  L-- 

dt 

Separating  the  variables,  we  have 

(49)  f  =  -f*' 

Integrating  both  sides  of  this  equation  between  the  limits 
zero  and  t,  at  which  instants-  the  currents  are  i0  and  i  respec- 
tively, we  have 

ff— |f4 

*/io   i  LJv 

whence 

(51) 


196  ELECTROMAGNETIC   INDUCTION     [VI,  §  138 

or 

(52)  log  !=_:?*, 

IQ  Ll 

or 

(53)  t  =  t0<r»"'. 

From  equation  (53)  it  is  seen  that  when  t  =  0,  i  has  the  value 
i0,  and  when  t  is  infinite,  i  becomes  zero.  The  graphical  rela- 
tion between  *  and  t  is  shown  in  curve  II,  Fig.  90.  For  a  time 
equal  to  L/ R  seconds,  it  is  seen  that 

(54)  i  =  !,:„, 

and  this  value  of  the  time,  in  which  the  current  falls  to  1/e 
times  its  initial  value,  is  called  the  time  constant.  This  time 
constant  is  a  characteristic  of  the  circuit,  by  means  of  which  we 
can  express  the  effect  of  the  inductance  in  retarding  the  fall 
of  the  current  to  zero. 

If  we  integrate  both  sides  of  equation  (53)  between  the 
limits  zero  and  infinity,  we  find  the  total  charge  Q  which  flows 
through  the  circuit  when  the  electromotive  force  is  cut  off : 

Q  =  C  i  dt  =  C  i0e~  *"Ldt ; 
whence 

(55)  Q='»!' 

From  equation  (55)  the  time  constant  is  given  a  new  signifi- 
cance ;  it  is  seen  to  be  that  time  interval  during  which  the 
initial  current  would  have  to  flow  if  it  remained  constant,  in 
order  to  convey  the  charge  which  does  actually  pass. 

In  the  design  of  electromagnets,  where  quick  action  is 
required,  or  where  the  time  relations  of  the  mechanism  to 
other  parts  is  specified,  the  time  constant  is  important,  as  it 
is  the  factor  which  controls  the  rate  of  rise  or  fall  of  the 
magnetizing  current. 


VI,  §  139]     CURRENT,   ENERGY,   AND  CHARGE          197 

For  an  average  telegraph  sounder  the  time  constant  will 
be  less  than  0.01  second.  For  a  relay  of  150  ohms  resistance, 
it  will  be  about  0.03  second.  For  the  field  magnets  of  a 
dynamo  it  may  be  as  high  as  ten  seconds.  These  figures 
signify  the  time  interval  required  for  a  growing  current  to 
reach  a  value  equal  to  0.63  of  the  final  value,  or  for  a  decreas- 
ing current  to  fall  to  0.37  of  its  initial  value. 

139.  The  Energy  in  an  Inductive  Circuit.  When  a  poten- 
tial difference  V  is  impressed  on  a  circuit  and  a  current  i  is 
established,  the  energy  dW  given  to  the  circuit  in  a  time  dt  is 
expressed  by  the  equation 

(56)  dW=  Vidt. 

If  the  circuit  is  inductive,  a  part  of  the  energy  is  stored  in 
the  magnetic  field  about  the  inductance,  while  another  part  is 
dissipated  as  heat  in  the  ohmic  resistance.  Separating  these 
components  and  remembering  that  L  (di/dt)  is  a  potential  differ- 
ence, we  may  write  equation  (56)  in  the  form 

(57)  dW  =  VR  dt+L- i  dt. 

dt 

In  order  to  find  the  energy  WL  stored  in  the  inductive  part  of 
the  circuit,  the  last  term  of  equation  (57)  may  be  integrated 
between  the  limits  zero  and  /,  where  /  represents  the  final 
value  of  the  current : 

(58)  WL 

This  is  called  the  intrinsic  energy  equation;  it  gives  the  value 
of  the  energy  stored  in  the  circuit  due  to  the  establishment 
of  the  magnetic  field.  If  the  current  is  reduced  to  zero,  the 
magnetic  field  will  collapse  and  disappear,  and  the  energy  will 
be  returned  to  the  circuit,  appearing  as  a  flow  of  charge,  if  the 
circuit  is  closed. 


198 


ELECTROMAGNETIC   INDUCTION     [VI,  §  139 


It  is  of  interest  to  note  the  analogy  between  equation  (58) 
and  the  expression  for  the  kinetic  energy  of  a  rotating 
body : 

K.E.  = 


where  K  is  the  moment  of  inertia  and  w  is  the  angular  speed. 
It  is  apparent  that  self-inductance  in  a  circuit  bears  the  same 
relation  to  a  changing  current  that  inertia  does  to  changing 
speed.  The  inductance,  like  the  inertia,  always  tends  to  op- 
pose the  change. 

When  a  hot  spark  is  required  for  ignition  purposes,  it  is 
not  sufficient  merely  to  break  the  circuit  of  a  dry  cell  or 
storage  battery,  for  the  potential  difference  appearing  at  the 
terminals  is  that  of  the  battery  only,  and  represents  but  a 
small  amount  of  power.  If,  however,  a  coil  of  large  induc- 
tance is  put  in  series  with  the  battery,  an  amount  of  energy 
given  by  equation  (58)  is  stored  in  the  magnetic  field  at  a 
relatively  slow  rate.  This  energy  may  be  made  as  large  as 
desired  by  the  "proper  regulation  of  L  and  /.  The  sudden 
breaking  of  this  circuit  causes  a  rapid  decrease  in  the  current, 
and  a  correspondingly  large  potential  difference  appears  at 
the  separated  terminals.  The  stored  energy  is  released 
quickly  and  a  hot  spark  is  the  result. 

140.  Inductance  and  Capacity  in  Parallel.  If  an  inductive 
coil  is  shunted  by  a  condenser  in  series  with  some  resistance, 
as  shown  in  Fig.  91,  the  time  constants  of  the  two  portions  of 

the  circuit  may  be  so  chosen 
that  the  effects  of  capacity 
and  inductance  will  annul 
each  other.  From  equa- 
tion (21),  §  125,  the  current 
flowing  through  A  A  alone 
91.  is  given  by  the  equation 


VI,  §  141]     CURRENT,  ENERGY,  AND  CHARGE          199 

(59)  ie  =  Ie- t/BC. 

From  equation  (46),  the  current  flowing  through  BB'  alone  is 
given  by  the  equation 

(60)  ^  =  /(l-e-^). 

If  the  time  constants  are  made  equal,  that  is,  if 

(61)  |  =  JRC, 

and  if  both  branches  of  the  circuit  have  the  same  potential 
difference  simultaneously  impressed,  then  the  total  current 
flowing  through  the  circuit  is  given  by  the  equation 

(62)  ie  +  iL  =  L 

This  equation  shows  that  under  the  assumed  conditions,  the 
current  which  flows  is  precisely  that  which  would  flow  if  the 
circuit  had  a  resistance  R  but  neither  capacity  nor  inductance. 
This  principle  is  applied  in  an  important  way  in  modern 
telephone  circuits,  where  the  inevitable  capacity  of  lines  and 
cables  distorts  the  wave  form  of  the  voice  currents,  thus 
greatly  diminishing  the  clearness  of  the  speech  transmission. 
The  effect  of  the  capacity  is  to  cause  the  various  components 
of  the  voice  currents  to  travel  with  different  speeds  and  to 
die  away  at  different  rates,  and  it  is  desirable  to  overcome  this 
effect  as  far  as  possible.  This  is  accomplished  by  introducing 
inductance  in  the  form  of  loading  coils  at  intervals  through- 
out the  circuit,  and  it  is  only  by  this  means  that  the  recent 
advances  in  long  distance  and  cable  transmission  have  been 
possible. 

141.  A  Relation  between  Self  and  Mutual  Inductance. 

Consider  two  circuits,  A  and  B,  connected  in  helping  series, 
with  the  same  current  i  flowing  in  each.  Let  La  and  Lb  repre- 
sent respectively  the  inductance  in  each  circuit,  and  let  M  be 


200  ELECTROMAGNETIC   INDUCTION     [VI,  §  141 

their  mutual  inductance.  By  equations  (8)  and  (22)  the  num- 
ber of  linkings  is 

(63)  for  the  A  circuit :  NA  =  LAi  +  Mi, 

(64)  for  the  B  circuit :  NB  =  LBi  +  ML 

Equation  (63)  gives  the  number  of  linkings  between  lines  of 
force  arising  in  the  A  circuit,  and  wire  turns  of  the  B  circuit. 
Equation  (64)  gives  the  number  of  linkings  between  lines  of 
force  arising  in  the  B  circuit,  and  wire  turns  of  the  A  circuit. 
The  total  number  N  of  linkings  due  to  lines  of  force  common 
to  both  circuits  is  then  given  by  the  sum  of  NA  and  NB\ 
whence 

(65)  N=NA  +  Nt  =  (LA  +  LB  +  2  M)i. 

If  A  represents  the  self-inductance  of  the  system  considered 
as  a  single  coil,  then 

N=Lli; 

whence 

(66)  A=(A<  +  A^2Jlf). 

If  the  coils  A  and  B  are  now  connected  in  opposing  series, 
the  method  applied  in  deriving  (63)  and  (64)  will  give, 

(67)  for  the  A  circuit :  NA  =  LAi  -  Mi, 

(68)  for  the  B  circuit :  NB  =  LBi  -  Mi. 

The  total  number  N  of  linkings  due  to  lines  of  force  common 
to  both  coils  will  be  given  by  the  sum  of  (67)  and  (68) ; 
whence 

(69)  N=NA  +  NB  =  (LA  +  LB-2  M)i. 

If  A  represents  the  self-inductance  of  the  system  considered 
as  a  single  coil,  we  have 

N=  L2i, 
whence 


VI,  §  142]     CURRENT,   ENERGY,   AND  CHARGE          201 
Subtracting  (70)  from  (66),  and  solving  for  M,  we  have 

(71)  M. 

If  two  similar  circuits  with  equal  self-inductances  are 
imagined  to  be  absolutely  superposed  in  space,  it  may  be 
shown  that 

(72)  LA  =  LB  =  M. 

142.  The  Quantity  of  Electricity  in  an  Inductive  Circuit. 

If  a  potential  difference  is  induced  in  a  circuit  by  some  change 
in  the  number  of  linkings  of  its  wire  turns  with  magnetic 
lines  of  force,  the  value  of  the  resulting  current,  in  general, 
will  not  be  constant.  Moreover,  the  induced  current  is  tran- 
sient, persisting  only  while  the  change  in  the  linkings  is  tak- 
ing place.  The  value  of  this  transient  and  variable  current  is 
difficult  to  measure.  In  general,  it  is  more  useful  to  measure 
the  total  quantity  Q  of  charge  which  passes.  The  value  of 
this  quantity  is  easily  calculated.  At  any  instant,  the  value 
of  the  current  flowing  through  a  resistance  R  under  a  poten- 
tial difference  Fis,  by  Ohm's  law, 

(73)  ||       m      «-£     '  .  H 

Remembering  that  i  =  dQ/dt,  and  that  V=dN/dt,  we  may 
write 

(74)  dQ  =  ±dN. 

Integrating  this  expression  for  dQ  between  the  limits  N2  an(i 
Nlt  which  are  respectively  the  initial  and  final  values  of  the 
flux  turns,  corresponding  to  the  values  zero  and  Q  of  the 
charge,  we  have 


202  ELECTROMAGNETIC   INDUCTION     [VI,  §  142 

Representing  the  change  in  flux  turns  by  AJV,  equation  (75) 
may  be  written  in  the  form 


This  equation  states  that  the  induced  charge  is  numerically 
equal  to  the  ratio  between  the  total  change  in  the  number  of 
linkings  and  the  total  resistance  of  the  circuit  through  which 
the  charge  passes.  It  is  seen  that  the  amount  of  the  induced 
charge  is  independent  of  the  time.  It  must  be  remembered 
that  the  mere  condition  that  lines  of  force  are  interlinked 
with  wire  turns  does  not  induce  the  flow  of  a  charge.  Induc- 
tion currents  only  arise  when  some  change  in  the  number  of 
linkings  occurs,  that  is,  when  the  lines  of  force  are  being  cut 
by  wire  turns. 

As  an  illustration,  consider  the  case  of  the  current  inductor 
(§  131),  with  a  steady  current  through  the  primary.  Let  the 
corresponding  number  of  linkings  be  denoted  by  JVj.  If  the 
circuit  is  broken,  the  magnetic  field  collapses,  and  N2  is  zero. 
For  this  case,  the  charge  induced  in  the  secondary  is 

fc  =  S. 

R 

If  current  is  started  through  the  primary,  the  linkings  with 
the  secondary  circuit  increase  from  zero  to  JV2,  and 


If  the  primary  current  is  reversed  in  direction,  the  lines  of 
force  are  first  withdrawn  and  then  immediately  reestablished 
in  the  opposite  direction.  For  this  case 


and 

(77) 


VI,  §  142]     CURRENT,   ENERGY,   AND  CHARGE          203 

The  quantity  is  given  in  absolute  units  when  N  and  R  are 
in  absolute  units.     If  R  is  in  ohms  and  Q  is  in  coulombs,  then 

(78)  Q:     2N 


EXERCISES 

1.  Check  the  equation  of  §  135  by  means  of  dimensional  formulas. 

2.  Check  the  denning  equations  for  self  and  mutual  inductance  by 
means  of  dimensional  formulas. 

3.  A  pair  of  circuits  has  a  mutual  inductance  of  3  millihenrys.    A 
current  of  1  ampere  is  started  through  the  primary.     What  is  the  value 
of  the  charge  induced  in  the  secondary,  if  its  resistance  is  0. 1  ohm  ? 

4.  A  circuit  has  a  resistance  of  1.0  ohm,  and  an  inductance  of  0.5 
henry.     An  initial  current  of  100  amperes  is  flowing,  when  the  impressed 
potential  difference  is  cut  off  by  opening  a  switch.     Find  (a)  the  initial 
rate  of  decrease  of  the  current ;  (6)  the  time  constant  of  the  circuit ; 

(c)  the  value  of  the  current  strength  for  t  =  0.01,  0.1,  and  1.0  second; 

(d)  the  rate  of  change  of  current  for  the  times  given  in  (c). 

5.  A  potential  difference  of  110  volts  is  impressed  on  a  circuit  of  re- 
sistance 3  ohms  and  inductance  of  0.04  henry.     Calculate  (a)  the  initial 
'rate  of  rise  of  current  strength ;  (6)  the  value  of  the  current  strength  for 
t  =  0.02  second ;  (c)  the  iE  drop  and  the  inductive  drop  at  this  instant ; 
(d)  the  rate  of  change  of  current  at  this  instant ;  (e)  make  the  calcula- 
tions of  (c)  and  (d)  for  t  =  0.5  second. 

6.  In  order  to  trace  the  rate  of  growth  of  a  current  in  an  inductive  cir- 
cuit, find  the  fraction  of  its  final  value  for  t  =  L/R,  t  =  2  L/R,  t  =  3  L/R, 
etc. 


CHAPTER   VII 

ELECTRICAL    QUANTITY   AND    THE   BALLISTIC 
GALVANOMETER 

143.  Fundamental  Relations.  In  the  preceding  chapters 
three  important  equations  have  been  discussed,  which  deal 
with  the  quantity  or  amount  of  charge  passing  through  a 
circuit. 

In  §  102,  it  was  shown  that  the  value  of  the  quantity,  when 
a  constant  current  i  flows  through  a  circuit  for  t  seconds,  is 
given  by  the  equation 

(1)  Q=t«; 

whereas,  if  the  current  is  not  constant,  the  value  of  the  charge 
is,  by  equation  (1),  §  117, 

(2)  Q 

In  §  117,  it  was  shown  that  the  quantity  stored  in  a  con- 
denser is  given  by  the  equation 

(3)  Q  =  CV. 

In  §  142,  equation  (76)  gives  the  value  of  the  total  induced 
charge  in  terms  of  the  change  in  the  number  of  linkings  and 
the  resistance  of  the  circuit,  in  the  form 

(4)  « 

T 

For  the  measurement  of  quantity  passing  more  or  less  con- 
tinuously through  a  circuit,  as  in  (1)  above,  and  of  relatively 

204 


VII,  §  144]     THE  BALLISTIC  GALVANOMETER  205 

large  value,  as  in  the  case  of  lighting  or  power  circuits,  a 
quantity  meter  in  some  form  must  be  used.  These  meters  are 
called  coulomb  meters  or  ampere-hour  meters.  For  a  detailed 
description  of  them  the  student  is  referred  to  the  larger  trea- 
tises and  the  journals. 

For  the  measurement  of  the  transient  and  relatively  small 
quantities  produced  by  the  discharge  of  a  condenser,  or  arising 
in  an  inductive  circuit,  as  in  (3)  and  (4)  above,  the  ballistic 
galvanometer  is  used. 

144.  The  Ballistic  Galvanometer.  The  term  "  ballistic  "  is 
derived  from  a  Greek  word  signifying  to  throw.  Its  descrip- 
tive meaning  is  obvious  when  it  is  considered  that  this  galva- 
nometer is  used  only  to  measure  time  integrals  of  transient 
currents.  The  conditions  under  which  it  is  used  do  not  admit 
of  steady  deflections,  but  rather  of  an  initial  throw,  followed 
by  a  series  of  swings  of  decreasing  amplitude.  It  is  this  first 
throw  which  is  observed,  and  which  is  found  to  be  proportional 
to  (and  therefore  a  measure  of)  the  passing  charge. 

The  current  through  the  galvanometer  is  not  constant,  and 
it  is  of  very  short  duration.  The  reaction  of  its  magnetic  field 
with  the  field  of  the  instrument  produces  an  impulsive  torque 
on  the  suspended  system.  This  is  opposed  by  the  torque  due 
to  (a)  the  elasticity  of  the  suspension  in  the  moving-coil  type, 
(6)  the  controlling  magnetic  field  in  the  moving-needle  type, 
and  (c)  the  torque  due  to  air  friction,  or  to  the  magnetic  field 
reactions  of  induced  currents,  set  up  either  in  metallic  parts  of 
the  instrument  or  in  the  wires  of  the  circuit  itself. 

The  moving  system  must  have  a  sufficient  moment  of  inertia 
so  that  it  does  not  begin  to  move  until  practically  all  the 
charge  has  passed,  and  its  motion  must  not  be  too  fast,  or  the 
observer  will  have  difficulty  in  reading  the  throw.  Any  gal- 
vanometer of  either  type  may  be  used  ballistically  with  certain 
restrictions. 


206  ELECTRICAL   QUANTITY  [VII,  §  145 

145.  The  Suspended-needle  Type.     The  moving-needle  bal- 
listic galvanometer   is   quite  like  any  other   sensitive   galva- 
nometer of   its  type  with  the  addition  of   two  characteristic 
features :  (a)  the  needle  is  massive,  or  in  some  way  loaded  so 
as  to  give  it  a  large  moment  of  inertia ;  (b)  the  galvanometer 
is  so  designed  that  the  damping  is  very  small. 

It  is  usually  made  with  a  small  bell-shaped  needle  in  order 
that  air  friction,  the  chief  cause  of  damping  in  this  type,  may  be 
avoided.  Its  sensibility  can  be  increased  by  using  the  astatic 
system,  and  may  be  varied  by  using  interchangeable  coils,  vari- 
able coil  distance,  or  variable  control  with  permanent  magnets. 

To  set  up  such  a  galvanometer,  select  coils  of  suitable  re- 
sistance, and  with  the  control  magnet  removed,  fix  the  mirror 
on  the  suspension  so  that  it  faces  away  from  the  scale.  Now 
by  means  of  the  control  magnet,  or  suitable  bar  magnets  laid 
flat  on  the  table  near  the  galvanometer,  bring  the  needle  about 
until  the  mirror  faces  the  scale.  Careful  manipulation  of  the 
magnets  will  give  a  very  small  residual  control ;  hence  the 
galvanometer  can  be  made  as  sensitive  as  desired.  A  working 
period  of  10  seconds  is  usually  sufficiently  sensitive,  while  a 
period  of  15  seconds  or  more  is  impracticable  under  ordinary 
working  conditions  because  of  stray  fields  from  adjacent  elec- 
tric circuits.  On  account  of  the  susceptibility  of  this  type 
of  galvanometer  to  such  disturbances,  it  has  been  largely 
superseded  by  the  moving-coil  instruments. 

For  a  suspended-needle  galvanometer,  if  damping  is 
neglected,  it  may  be  shown  that  the  relation  between  the 
charge  passing  through  its  coils  and  the  deflection  is  given  by 

(5)  Q=0sin|, 

where  G  is  a  constant  and  d  is  the  first  throw. 

146.  The  Suspended-coil  Type.     In  the  moving-coil  type 
of  ballistic  galvanometer,  the  transient  current  sets  up  a  mag- 


VII,  §  147]     THE  BALLISTIC   GALVANOMETER  207 

netic  field  about  the  coil  which  lasts  only  as  long  as  the  cur- 
rent flows.  This  field,  during  the  brief  time  for  which  it 
exists,  reacts  with  the  field  of  the  permanent  magnet,  and  this 
causes  an  impulsive  torque  and  a  sudden  angular  displacement 
of  the  system.  The  relation  between  the  quantity  of  elec- 
tricity passing  and  the  resulting  deflection  is  given  by  the 
equation 

(6)  Q=Gd, 

where  G  is  a  constant  and  d  is  the  first  throw. 

The  values  of  the  constants  in  equations  (5)  and  (6)  can  be 
determined  readily  from  the  theory,  but  the  calculations  in- 
volve an  accurate  knowledge  either  of  the  dimensions  of  the 
parts  and  the  magnetic  field  strength,  or  of  the  figure  of  merit 
and  the  period  ;  and  the  result  must  be  corrected  for  damping. 
Practically  such  calculations  are  of  little  value.  Methods  for 
finding  these  constants  experimentally  will  be  discussed  in 
§§  149-151. 

147.  Damping.  A  simple  pendulum  vibrating  in  air  will 
make  a  great  many  vibrations  before  it  finally  comes  to  rest. 
If  the  pendulum  ball  swings  in  oil  or  water,  its  motion  will  be 
retarded  to  a  much  greater  degree,  and  the  vibrations  will 
cease  much  sooner  than  before.  Indeed  the  medium  may  have 
sa  great  a  viscosity  that  the  pendulum,  after  deflection  and 
release,  will  return  slowly  to  its  equilibrium  position  without 
crossing  the  zero  point.  The  same  variations  in  the  motion 
of  the  suspended  system  of  a  galvanometer  may  be  observed  if 
the  damping  is  varied  through  wide  limits.  * 

By  damping  is  meant  the  effect  of  all  the  retarding  forces 
which  oppose  the  motion  of  a  vibrating  body.  By  means  of 
these  forces,  the  energy  stored  in  the  body  is  gradually  dissi- 
pated. It  is  usually  assumed  that  the  resultant  of  these  forces 
is  proportional  at  every  instant  to  the  velocity  of  the  body. 


208 


ELECTRICAL  QUANTITY 


[VII,  §  147 


Air  friction  causes  damping  in  either  type  of  galvanometer. 
In  the  moving-coil  type  a  more  effective  cause  of  damping  is 
the  establishment  of  induced  currents  in  the  metal  parts  of 
the  coil  frame,  or  in  the  circuit  of  the  coil  itself  when  the  ter- 


10  15 

Seconds 
FIG.  92. 

minals  are  connected.     Frequently  a  small  closed  rectangle  of 
copper  wire  is  attached  to  the  coil  for  this  purpose. 

It  may.  be  shown .  from  the  general  equations  for  oscillatory 
motion,  that  for  a  strictly  undamped  condition,  the  deflection 
is  a  simple  harmonic  function  of  the  time,  of  the  form 

(7)  d  =  k  sin  bt, 

where  bt  is  the  phase  angle.     The  graphic  representation  of 


VII,  §  148]     THE  BALLISTIC  GALVANOMETER  209 

this  relation  between  deflection  and  time  is  shown  in  curve  1, 
Fig.  92.  If  the  damping  is  small,  the  successive  amplitudes 
slowly  decrease  in  value,  as  shown  in  curve  2.  As  the  damp- 
ing increases,  the  suspended  system  comes  to  rest  more 
quickly,  as  shown  in  curve  3.  For  damping  greater  than  a 
certain  critical  value,  the  motion  loses  its  periodic  or  oscillatory 
character  and  follows  curve  4. 

This  latter  type  of  non-oscillatory  motion  is  called  aperiodic. 
It  is  desirable  in  many  forms  of  measuring  instruments,  inas- 
much as  the  observer  does  not  have  to  wait  for  the  oscillatory 
motion  to  die  away. 

Moving-coil  galvanometers  may  be  so  designed  that  the  first 
throw  is  proportional  to  the  quantity  passing  through  •  the 
coils,  even  though  the  damping  is  large  enough  to  produce 
the  dead  beat  or  non-oscillatory  motion.  The  sensibility  is 
greatly  reduced,  however.  A  Weston  voltmeter,  or  indeed 
any  galvanometer  in  which  the  damping  is  chiefly  due  to  in- 
duced currents,  will  give  a  straight-line  curve  when  charges 
are  plotted  against  throws.  If  the  induced  currents  flow 
through  the  circuit  connected  to  the  galvanometer  terminals, 
the  damping  will  vary  with  the  resistance  of  the  circuit,  that 
is,  the  factor  G  in  equation  (6)  will  not  be  a  constant  for  all 
values  of  the  circuit  resistance. 

148.  Logarithmic  Decrement.  On  account  of  the  damping, 
the  alternate  amplitudes  of  swing  of  a  galvanometer  on  either 
side  of  the  equilibrium  position  grow  successively  less.  If 
dit  d2,  etc.,  represent  these  successive  amplitudes,  we  may  write 

(8)  £_J  =  J=  ...-*. 

«2        "3        tt4 

The  factor  k  is  called  the  damping  factor,  or  the  damping  ratio, 
and  its  logarithm  to  the  base  e  is  called  the  logarithmic 
decrement,  which  is  usually  represented  by  A..  Any  deflection 


210  ELECTRICAL  QUANTITY  [VII,  §  148 

is  less  than  it  would  have  been  if  there  had  been  no  damping. 
If  d0  represents  the  throw  which  would  have  occurred  if  the 
motion  had  been  entirely  undamped,  and  d  the  observed  first 
throw,  it  may  be  shown  that  we  have  approximately  1 

(9)  *= 

or 

(10)  dQ  = 

These  equations  must  be  regarded  as  giving  approximate  rela- 
tions only,  and  the  method  loses  its  validity  when  the  correc- 
tion factor  reaches  a  value  larger  than  about  1.20. 

The  theory  of  logarithmic  decrement,  with  methods  of 
determining  values  of  the  correction  factors,  was  formerly  im- 
portant when  using  suspended  needle  galvanometers.  Pres- 
ent practice  employs  almost  exclusively  the  suspended  coil 
type,  rather  heavily  damped,  for  which  such  correction  factors 
are  of  no  significance.  Logarithmic  decrement  and  damping 
are  of  great  importance,  however,  when  dealing  with  oscilla- 
tory currents,  such  as  are  constantly  used  in  high  frequency 
circuits  and  wireless  telegraphy,  and  a  full  discussion  of  the 
theory  will  be  found  in  the  larger  works  upon  those  subjects. 

149.  Calibration  of  a  Ballistic  Galvanometer.  The  cali- 
bration of  a  ballistic  galvanometer  consists  in  sending  known 
charges  of  electricity  through  the  instrument,  and  observing 
the  corresponding  deflections.  These  charges  may  be  produced 
in  either  of  two  ways,  as  follows  : 

I.  With  a  standard  condenser  and  a  battery  of  known 
electromotive  force,  the  charge  stored  in  the  condenser  is  given 
by  equation  (2),  §  117, 

1  Theoretically,  the  approximate  relation  is  d=  do  •  e~^/2or  d0=d'e^/2. 

Since  e*/2  --=  1  +  X/2  H ,   we  have  approximately   the   relation  (9).     Since 

*A  =  k,  we  have  e^/2  =  V&,  whence  d0  =  d  •  Vjfc,  as  in  (10). 


VII,  §  149]     THE  BALLISTIC   GALVANOMETER 


211 


(11)  Q  =  CV. 

The  galvanometer  constant  is   given  by  equation  (6)  in  the 

form 


d       d 

II.  By  means  of  any  arrangement  of  a  magnetic  field  and 
of  wire  turns  of  a  circuit,  in  which  the  change  in  the  number 
of  linkings  can  be  computed,  we  can  find  the  charge  from  the 
relation  given  in  equation  (76),  §  142, 


(12) 


R 


Such  an  arrangement  can  be  realized  by  means  of  a  mutual 
inductance  M,  connected  as  shown  in  Fig.  93.  Any  change  in 
the  current  through  the 
primary  coil  P  is  accom- 
panied by  an  induced 
charge  in  the  secondary 
circuit  which  causes  a 
throw  on  the  ballistic 
galvanometer. 

The  value  of  the 
charge  is  derived  as 
follows.  A  constant 

battery  of  suitable  electromotive  force  is  put  in  series  with  a 
control  rheostat  R',  an  ammeter  Am,  and  a  reversing  switch 
W.  The  switch  W  enables  us  to  reverse  the  current  through 
the  primary  of  the  mutual  inductance.  The  secondary  coil  S 
is  in  series  with  the  galvanometer  BO,  and  with  an  adjustable 
resistance  r.  If  the  current  strength  in  the  primary  coil  P  is 
changed  from  some  initial  value  ii  to  some  final  value  i2,  a 
corresponding  change  takes  place  in  the  number  of  linkings 
with  the  wire  turns  of  the  secondary  coil  S.  The  charge 
induced  is  given  by  equation  (75),  §  142,  in  the  form 


FIG.  93. 


212  ELECTRICAL  QUANTITY  [VII,  §  149 

(13)  Q_^LTlZ» 

where  R  is  the  total  resistance  of  the  secondary  circuit.     By 
equation  (8),  §  130,  this  equation  becomes 

(14)  Q  = 


R 

If  the  current  is  brought  from  zero  to  some  value  i2  by  clos- 
ing the  switch,  we  have 

(15)  «  —  IT 

If  the  initial  current  t\  is  broken  by  opening  the  switch,  i.2  =  0, 
and  we  have 


The  changed  sign  shows  that  the  induced  charge  is  in  oppo- 
site directions  in  these  two  cases.  If  the  initial  current  is 
reversed  by  throwing  over  the  switch  W,  then 

ni\  o  =  M&  -_(-  y  J  =  2J^'i 

R  R 

By  (6),  the  value  of  the  galvanometer  constant  is  given  by 

the  equation 

Q  =  Gd, 
whence 

(18)  Q  =  2-ji>  =  Gd, 

or 
(19) 


Rd 


In  equation  (18),  Q  will  be  in  coulombs  when  M  is  in  henry  s, 
R  in  ohms,  and  i  in  amperes. 

In  case  a  standard  current  inductor  is  used,  equation  (19) 
becomes,  by  (16),  §  131, 


VII,  §  149]     THE  BALLISTIC  GALVANOMETER 

2  ML      8 


213 


(20) 


It 


R 


This  gives  Q  in  absolute  units  if  i  and  R  are  in  absolute  units. 
To  get  Q  in  coulombs,  i  and  R  must  be  in  amperes  and  ohms 
respectively,  and  the  number  of  linkings  must  be  divided  by 

109 ;  this  gives 

x     8  7r2nr2  Si* 


After  several  pairs  of  corresponding  values  of  the  charge 
and  the  deflection  have  been  found,  they  may  be  plotted  on 
cross-section  paper.  If  G  is  constant,  for  any  particular  in- 
strument, the  curve  will  be  a  straight  line.  Figure  94  shows 
several  such  curves  for  a  galvanometer,  when  different  resist- 


1  eo 

2  1670  ohms 

3  1370 

4  1070 
6  770 
6  670 


Q 

FIG.  94. 

ances  are  connected  in  series  with  it.  The  values  for  Q  used 
in  plotting  curve  1  were  obtained  from  a  standard  condenser 
and  standard  cell.  (See  §  150.)  In  this  case  the  galvanometer 
was  in  series  with  a  resistance  of  some  hundreds  of  megohms, 
the  condition  being  practically  that  of  open  circuit,  so  far  as 
induced  currents  in  the  coil  are  concerned. 

The  values  of  Q  in  the  other  curves  were  obtained  from  a 


214  ELECTRICAL   QUANTITY  [VII,  §  149 

standard  of  mutual  inductance  by  the  method  of  §  151.  The 
values  of  the  total  secondary  circuit  resistances  are  given  for 
each  of  the  curves.  It  is  obvious  from  an  inspection  of  these 
calibration  curves  that  the  throw  for  any  given  value  of  Q  in  • 
creases  as  the  circuit  resistance  increases,  being  greatest  when 
the  condenser  method  is  used.  In  this  case  the  galvanometer 
is  swinging  with  its  circuit  practically  open,  so  that  the  throw 
is  not  influenced  by  damping  due  to  induced  currents. 

In  the  case  of  the  slightly  damped  galvanometer  systems, 
for  which  the  motion  is  oscillatory,  the  application  of  the  cor- 
rections for  damping  will  reduce  the  calibration  curves  taken 
under  different  conditions  of  circuit  resistance  to  a  standard 
curve  for  undamped  motion.  Moreover,  this  curve  will  be 
coincident  with  the  calibration  curve  by  using  a  standard  con- 
denser after  it  has  been  corrected  for  damping. 

In  the  case  of  the  heavily  damped,  or  dead  beat  moving 
coil  galvanometers,  there  is  no  simple  way  of  reducing  any 
calibration  curve  to  an  undamped  standard  curve,  and  it  be- 
comes necessary  to  calibrate  the  galvanometer  for  the  precise 
condition  under  which  it  is  to  be  used.1 

In  using  the  ballistic  galvanometer  it  is  desirable  that  the 
throw  shall  be  slow  enough  to  be  easily  read,  that  the  time  of 
return  shall  be  as  small  as  possible,  and  that  the  sensibility 
shall  be  as  great  as  possible.  It  has  been  shown  that  these 
conditions  are  best  fulfilled  when  the  damping  is  just  suffi- 
cient to  make  the  galvanometer  aperiodic.  This  condition  can 
usually  be  realized  by  the  adjustment  of  the  resistance  in 
series  with  the  instrument. 

1  Special  keys  have  been  devised  which  introduce  a  fixed  resistance  into 
the  galvanometer  circuit  immediately  after  the  condenser  charge  has  passed, 
or  which  open  the  circuit  immediately  after  the  inductive  charge  has  passed. 
Such  keys  are  easily  arranged  and  should  always  he  used  if  it  is  necessary  to 
calibrate  the  galvanometer  with  one  type  of  circuit  and  use  it  with  another 
type  of  circuit.  In  general,  however,  the  galvanometer  should  be  calibrated 
under  the  same  conditions  of  circuit  resistance  as  those  with  which  it  will  be 
used. 


VII,  §  150]     THE  BALLISTIC  GALVANOMETER 


215 


150.  Laboratory  Exercise  XXVII.  Calibration  of  a  moving 
coil  ballistic  galvanometer  with  a  standard  condenser. 

APPARATUS.  Standard  adjustable  condenser,  standard  cell, 
discharge  key,  and  ballistic  galvanometer. 

The  discharge  key,  shown  in  Fig.  95,  is  a  highly  insulated 


FIG.  95. 


three-way  key,  arranged  for  prompt  and  convenient  manipu- 
lation. The  binding  post  in  contact  with  the  rocking  lever 
is  always  attached  to  one  of  the  condenser  terminals.  The 
battery  is  put  in  series  with  the  condenser  through  the  upper 


FIG.  96. 


contact  points,  and  the  galvanometer  is  connected  to  the  con- 
denser through  the  lower  contact  points. 

PROCEDURE.     (1)    Arrange  the  circuit  as  in  Fig.  96.     With 
k  pressed  to  b,  the  condenser,  of  capacity  O,  is  charged  with 


216  ELECTRICAL  QUANTITY  [VII,  §  151 

the  potential  difference  V  due  to  the  standard  cell.  Then  the 
charge  Q  given  to  the  condenser  is  given  by  the  equation 

Q=CV. 

With  k  raised  to  a,  this  charge  is  sent  through  the  galvanom- 
eter, and  the  corresponding  deflection  is  observed.  The  value 
of  G  is  then  given  by  the  equation 


(2)  In   order  to  determine  whether  G   is  constant   for  all 
values  of  the  deflection  d,  it  is  necessary  to  vary  Q  and  take 


FIG.  9(i  (repeated). 

deflections  over  the  entire  working  range  of  the  scale.  This 
variation  in  Q  is  most  easily  secured  by  using  an  adjustable 
capacity  at  C.  In  case  a  condenser  of  fixed  value  only  is 
available,  a  storage  battery  and  volt  box  may  be  used  at  B, 
the  values  of  the  potential  differences  being  determined  with 
a  potentiometer  or  an  accurate  voltmeter. 

(3)  Take   several  readings  of  the  scale  deflection  for  each 
value  of  the  capacity,  and  record  the  E.  M.  F.  of  the  battery 
used. 

(4)  Calculate  the  value  of  Q  for  each  observed  value  of  d, 
and  plot  a  curve  coordinating  Q  in  microcoulombs  and  deflec- 
tions.    The  quantity  Q  .will  be  expressed  in  coulombs  when  Q 
is  in  farads  and  V  is  in  volts.     If  C  is  in  microfarads,  Q  will 


VII,  §  151]     THE  BALLISTIC  GALVANOMETER  217 

be  in  inicrocoulornbs.     The  observed  values  of  d  are  subject 
to  correction  for  damping. 

In  all  experiments  using  condensers,  it  is  essential  that  the  insulation 
of  the  different  parts  of  the  circuit  should  be  high.  The  crossing  of  wires 
and  contact  between  the  wires  and  the  table  should  be  avoided.  Con- 
siderable practice  is  necessary  in  order  to  observe  the  first  throw  accu- 
rately, owing  to  the  rapid  motion  of  the  galvanometer  coil. 

151.  Laboratory  Exercise  XXVIII.  Calibration  of  a  ballistic 
galvanometer  with  a  standard  mutual  inductance. 

APPARATUS.  Standard  mutual  inductance,  ballistic  galva- 
nometer, reversing  switch,  ammeter,  storage  battery,  and  con- 
trol rheostat. 

PROCEDURE.  (1)  Arrange  the  circuit  as  in  Fig.  93,  making 
r  equal  to  zero. 

(2)  Start  with  some  small  value  of  the  current,  reverse  W, 
and  note  the  deflection.  Reduce  Rf  until  the  deflection  on 
reversal  is  nearly  full  scale.  Read  and  record  several  values 
of  d  for  successive  reversals,  together  with  corresponding 
values  of  the  current.  Reduce  the  current  and  repeat  the 
readings  with  deflections  about  half  as  large  as  before. 

REPORT.  (1)  Tabulate  all  the  readings,  together  with  the 
means  of  the  galvanometer  deflections.  Record  also  the  total 
resistance  of  the  secondary  circuit,  and  the  probable  precision 
of  the  ammeter  and  galvanometer  readings. 

(2)  Plot  a  curve  between  microcoulombs  as  abscissas  and 
deflections  as  ordinates.     Account  for  the  form  of  the  curve 
if  it  departs  from  a  straight  line. 

(3)  In  order  to  show  the  variations  in  these  curves  plot  the 
calibration  curve  for  the  galvanometer  as  taken  by  the  method 
of  §  150.     On  the  same  sheet  plot  the  curve  as  found  above. 

Increase  r  from  zero  to  some  value  such  that  the  total  resist- 
ance of  the  secondary  circuit  is  made  successively  two,  three, 
and  four  times  the  original  value.  In  the  manner  described 
in  (2)  take  data  for  calibration  curves  for  these  three  condi- 


218 


ELECTRICAL  QUANTITY 


[VII,  §  151 


tions,  and  plot  them  also  on  the  same  sheet  with  the  other 
two.  Account  for  the  variation  in  the  slope  of  the  curves  and 
state  what  inferences  may  be  drawn  from  the  experiment. 

152.  The  Magneto-Inductor.     The   magneto-inductor  is  a 
convenient  and  self-contained  device  for  producing  a  change 

in  the  number  of  linkings 
between  the  lines  of  force 
of  a  fixed  bar  magnet  and 
the  wire  turns  of  a  movable 
coil.  With  this  change  in 
the  number  of  linkings 
known,  the  value  of  the 
induced  charge  in  a  circuit 
of  given  resistance  may  be 
found.  One  form  of  the 
apparatus  represented  in 
Fig.  97  consists  of  a  bowl- 
shaped  casting  ABC,  in  the  center  of  which  is  fixed  a  strong 
and  permanent  bar  magnet  NS.  Attached  to  the  top  of 
the  bar  magnet  is  a  circular  plate  which  almost  completes 
the  magnetic  circuit,  leaving  a  narrow  circular  air-gap  at  bb. 
Through  this  air-gap  passes  freely  a  brass  tube  DD,  which 
carries  at  cc  a  coil  of  fine  wire.  The  ends  of  this  wire  are 
attached  to  two  binding  posts  at  the  top  of  the  tube. 

Let  the  magnetic  flux  across  the  air-gap  be  represented  by 
<j>.     Then  with  S  turns  in  the  coil,  the  change  in  the  number 
of  linkings  as  the  coil  drops  freely  through  the  air-gap  is  given 
by  the  equation 
(22)  N=S<f>. 

If  R  is  the  total  resistance  of  the  circuit  which  includes  the 
galvanometer  and  the  coil, 'then  the  charge  induced  is 


D 

b 

& 

^c 

A 

' 

c 

N 

- 

D  . 

ti 

n 

FIG.  97. 


(23) 


g-f-cw, 


VII,  §  153]     THE  BALLISTIC  GALVANOMETER  219 

In  order  to  eliminate  G  and  determine  N  we  may  write,  by 

equation  (18), 

(24)  Q  =  2^ 


where  M  is  a  mutual  inductance  used  as  in  §  151.     Dividing 

(23)  by  (24)  and  solving  for  N,  we  have 


(25)  N 

The  total  flux  across  the  air-gap  is  found  from  the  equation 

(26)  §  f=*  =  ?Jf.       . 

If  A  is  the  area  of  cross-section  of  the  bar  magnet  NS,  the 
flux  density  through  it  is 


If  M  and  i  are  measured  in  henry  s  and  amperes,  respectively, 
the  equation  (25)  becomes 

(28)  N=2Mi-W*, 

and  the  same  factor,  108,  must  be  introduced  in  equation  (26) 
and  (27). 

If  the  magnet  has  been  properly  made,  and  if  the  air-gap  is 
small,  the  flux  lines  will  be  constant  in  number  for  a  long  time. 
The  device  then  affords  a  rapid  method  of  calibrating  a  ballis- 
tic galvanometer  with  a  minimum  of  apparatus.  If  used  in 
circuits  like  those  of  §§  222  and  236,  the  drop  coil  should  be 
left  connected  in  the  galvanometer  circuit  throughout  the  ex- 
periment in  order  to  keep  the  circuit  resistance  constant. 

153.  Laboratory  Exercise  XXIX.  To  determine  the  con- 
stants of  a  magneto-inductor. 

APPARATUS.  Magneto-inductor  and  apparatus  as  in  Labora- 
tory Experiment  XXVIII,  §  151. 


220  ELECTRICAL  QUANTITY  [VII,  §  153 

PROCEDURE.  (1)  Arrange  the  circuit  as  in  Fig.  93.  Con- 
nect the  coil  of  the  magneto-inductor  in  series  with  the  ballistic 
galvanometer  and  the  secondary  of  the  mutual  inductance. 
Drop  the  coil  and  observe  the  throw  d.  Repeat  several  times, 
using  different  values  of  S  if  possible. 

(2)  With  the  coil  fixed  in  position,  reverse  a  suitable  current 
through  P,  and  observe  the  corresponding  throw  D.     Repeat 
with  several  different  values  of  the  current  strength. 

(3)  From  these  data  calculate  the  value  of  N,  using  equation 
(28).     Also  find  values  of  <£  and  B,  and  state  the  units  in 
which  all  the  quantities  are  expressed.     Make  a  study  of  the 
apparatus  in  order  to  determine,  whether  the  drop  coil  is  cut- 
ting all  the  lines  of  force  due  to  the  magnet. 

EXERCISES 

1.  Given  a  magneto-inductor  together  with  its  constants,  and  a  ballistic 
galvanometer.     Show  how  the  ballistic  constant  of  the  galvanometer  is 
determined. 

2.  Show  fully  from  Lenz's  law  that  the  charge  induced  in  the  second- 
ary circuit  of  a  mutual  inductance  on  reversing  the  current  in  the  primary, 
is  twice  as  great  as  for  either  make  or  break. 


CHAPTER  VIII 
THE   MEASUREMENT   OF   CAPACITY 

154.  General  Considerations.  The  measurement  of  ca- 
pacity is  an  important  part  of  the  work  of  an  electrical  labora- 
tory. It  involves  the  calibration  of  standards,  the  comparison 
of  laboratory  or  secondary  standards,  and  measurements  on 
cables,  aerial  or  submarine,  and  telegraph,  telephone,  and 
power  lines. 

Capacity  is  of  special  importance  when  present  in  alternat- 
ing current  circuits.  For  an  alternate  charge  and  discharge  at 
low  frequencies,  such  as  those  used  in  commercial  light  and 
power  circuits,  capacity  measurements  agree  with  those  made 
by  single  charge  and  discharge. 

Under  the  action  of  high  frequencies  where  the  charges  and 
discharges  alternate  very  rapidly,  from  a  few  thousand  per 
second  upward,  a  given  capacity  will  measure  less  than  for  a 
single  prolonged  charge,  or  for  a  comparatively  slow  charge. 
For  the  theory  and  methods  of  capacity  measurement  at 
high  frequency  the  student  should  consult  the  textbooks  and 
treatises  on  wireless  telegraphy  and  high  frequency  circuits. 
Only  methods  which  are  applicable  to  low  frequency  or  to 
static  charge  and  discharge  will  be  considered  here. 

A  capacity  may  be  measured  by  comparison  with  a  standard 
capacity,  by  comparison  of  its  stored  charge  with  that  dis- 
charged by  a  self  or  mutual  inductance,  and  by  absolute  methods 
which  are  independent  of  previously  measured  standards. 

Suppose  two  condensers  of  capacities  C^  and  (72  are  charged 
respectively  with  potential  differences  V\  and  F2,  developing 

221 


222      THE  MEASUREMENT  OF  CAPACITY     [VIII,  §  154 

charges  Ql  and  Q2.     These  relations  may  be  expressed  by  the 
equations 

(1)  «,=  ClFi 

and 

/2\  Q  fl.V 

If   the   same   charge   is   given   to   both  condensers,  which  is 
always  the  case  if  they  are  connected  in  series,  we  have 

or 

(4)  VI=Y*. 

r      v 

^2     "\ 

If  the  two  condensers  are  charged  with  the  same  potential 
difference,  however,  we  have 

(5)  F!=F2; 

dividing  (1)  by  (2),  we  obtain  the  relation 

C2      Q.2     d2 

The  equations  (5)  and  (6)  give  two  fundamental  relations  by 
which  capacities  are  generally  compared.  In  equation  (4) 
potential  differences  may  be  measured,  or  better,  their  ratio 
may  be  found.  In  equation  (6)  a  ballistic  galvanometer  is 
required  to  measure  the  quantities. 

155.  Comparison  of  Capacities  by  Direct  Deflection.  Sup- 
pose ft  (Fig.  98)  represents  the  capacity  of  a  standard  con- 
denser and  ft  is  that  of  a  condenser  whose  capacity  is  to  be 
measured.  Either  condenser  may  be  introduced  separately 
into  the  circuit  by  means  of  the  three-way  key  k.  A  discharge 
key,  like  that  shown  in  Fig.  95,  is  placed  at  K,  and  serves  to 
charge  either  condenser,  and  to  discharge  it  through  the 
ballistic  galvanometer.  With  k  across  1  and  3,  and  with  K 


VIII,  §  155]     DIRECT  DEFLECTION  METHOD  223 

pressed  to  point  b,  the  charge  Qj  in  condenser  Ci  is  given  by 
the-  equation 


where  V  is  the  potential  difference  of  the  battery.     When  k  is 
raised  to  point  a,  this  charge  passes 
through  the  galvanometer  and  we 
have 

(7)  Q1=C1F=£c*1, 

where  G  is  the  ballistic  constant 

of  the  galvanometer.     If  k  is  now 

made  to  connect  the  points  2  and 

3,  and  K  is  thrown  in  succession  to  6  and  a  as  before,  the 

charge  in  C  is  given  by  the  equation 

(8)  Q2=C,V=Gd2. 
Dividing  (7)  by  (8),  we  obtain  the  relation 


whence 

(9)  Cf=C*. 

#! 

In  the  method  as  just  described,  the  galvanometer  is  in 
circuit  with  the  condenser  throughout  the  entire  time  of  the 
deflection.  During  this'  time  some  part  of  the  absorbed  charge 
is  certain  to  appear.  The  effect  of  this  absorption  will  be  to 
produce  a  current  in  the  galvanometer  coil  during  the  entire 
time  of  the  deflection.  This  tends  to  increase  the  deflection, 
and  the  longer  the  period  of  the  galvanometer,  the  greater 
becomes  the  effect  of  the  absorbed  charge.  This  leads  to  a 
value  of  the  capacity  which  increases  with  the  period  of  the 
galvanometer,  and  hence  it  is  a  variable  magnitude,  which 
depends  upon  the  experimental  conditions. 


224      THE  MEASUREMENT  OF  CAPACITY     [VIII,  §  155 


o.oi 


2.0 


6.0 


FIG.  99. 


Figure  99  shows  the  relation  between  the  quantity  dis- 
charged from  a  given  condenser  and  the  time.  Assume  that 
during  the  first  0.01  second  practically  all  of  the  charge  has 
passed,  and  that  the  absorbed  charge  does  not  all  appear  until 
some  seconds  later.  If  now,  the  condenser  can  be  disconnected 
from  the  galvanometer  immediately  after  the  point  n  is  reached, 

then  the  quantity  sent  through 
the  coils  will  be.  independent 
of  the  absorbed  charge,  and 
hence  of  the  period.  The 
charge  which  passes  before 
the  point  n  is  reached  may 
be  called  the  free  charge,  and 
the  capacity  computed  there- 
from may  be  called  the  free 
capacity.  Standard  condensers 
thus  rated  may  be  measured  with  a  high  degree  of  precision, 
provided  the  temperature  coefficient  of  the  condenser  is  known 
and  the  corresponding  correction  is  applied.  To  compare  the 
free  capacities  of  two  condensers,  a  discharge  key  is  required 
which  will  open  the  galvanometer  circuit  within  a  few  hun- 
dredths  of  a  second  after  the  battery  is  applied. 

156.  Laboratory  Exercise  XXX.  To  compare  capacities  by 
the  direct  deflection  method. 

APPARATUS.  Standard  condenser,  the  condenser  to  be  meas- 
ured, ballistic  galvanometer,  three-way  key,  discharge  key,  and 
one  or  more  battery  cells. 

PROCEDURE.  (1)  Arrange  the  circuit  as  in  Fig.  98.  Set  the 
three-way  key  on  the  standard  condenser  side,  and  charge  the 
condenser.  Then  quickly  discharge  through  the  galvanometer 
and  read  the  corresponding  throw.  This  should  be  repeated 
several  times.  The  mean  of  these  readings  may  be  called  d^ 

(2)  With  the  three-way  switch  on  the  unknown  condenser 


VIII,  §  157]     DIRECT   DEFLECTION   METHOD  225 

side,  a  similar  set  of  readings  will  be  taken.     The  mean  of 
these  may  be  called  dz. 

(3)  Calculate  from  equation  (9)  the  value  of  the  unknown 
capacity. 

(4)  The  readings  may  be  repeated  with  different  values  of 
V,  thus  bringing  the  deflection  to  different  parts  of  the  scale. 

The  chief  objection  to  the  method  is  that  deflections  must 
be  observed.  Zero  methods  are  always  preferable  when 
possible. 

In  case  widely  different  values  of  capacity  have  to  be  com- 
pared, it  is  well  to  vary  the  values  of  V  so  that  the  deflections 
may  be  kept  approximately  the  same.  This  "can  be  done  by 
using  a  volt  box  in  series  with  the  battery,  the  charging  poten- 
tials being  taken  from  traveling  contacts  AB  (Fig.  63). 

Since  the  voltage  between  A  and  B  is  proportional  to  the 
resistance  between  these  points,  equation  (9)  may  be  written 
in  the  form 

(10)  ft=C1^. 

r2«i 

It  is  important  that  the  insulation  should  be  high.  For  this  reason, 
all  crossing  of  wires  or  contact  of  the  insulated  covering  with  the  table 
or  other  pieces  of  apparatus  should  be  avoided.  The  connecting  wires 
should  be  run  as  air  lines  throughout.  This  precaution  will  be  necessary 
in  all  work  on  capacity. 

157.  Laboratory  Exercise  XXXI.  To  study  leakage,  absorp- 
tion and  residual  charge. 

APPARATUS.  Condenser,  ballistic  galvanometer,  discharge 
key,  and  one  or  more  cells  of  battery. 

PROCEDURE.  I.  Leakage.  (1)  Arrange  the  circuit  as  in 
Fig.  96.  Charge  C  by  pressing  k  to  b  for  an  instant,  and 
immediately  discharge  through  the  galvanometer,  reading  the 
deflection  d^  Again  charge  (7,  insulate  by  holding  k  on  the 
middle  position,  and  after  10  minutes  discharge,  read  the 
throw  d. 


226     THE  MEASUREMENT  OF  CAPACITY     [VIII,  §  157 

(2)  Continue  in  this  way,  making  the  time  for  which  the 
key  is  on  the  middle   position  successively  longer  until    the 
discharge  throw  has  reached   about   one   tenth   of   its  initial 
value. 

(3)  Plot  a  curve  between  Values  of  time  and  throw.     Since 
the  throws  are  proportional  to  the  charges,  the  curve  may  be 
used  to  calculate  the  percentage  of  the  original  charge  which 
leaks  away  per  minute.     For  a  given  interval,  say  the  first  one 
of  ten  minutes,  the  leakage  is  proportional  to  c?:  —  d2,  while 
the  percentage  that  leaks  away  per  minute  is  given   by  the 

formula 

d-d       100 


(4)  If  the  leakage  is  small,  as  it  is  in  a  mica  condenser,  the 
time  interval  for  which  k  is  held  on  the  middle  point  is  taken 
longer.  If  the  leakage  is  rapid,  as  it  is  in  a  paraffined-paper 
condenser,  the  interval  may  be  a  few  seconds  only. 

II.  Absorption.     (1)  With  the  circuit  as  in  Fig.  96,  and  a 
mica  condenser  at  O,  charge  and  immediately  discharge  through 
the  galvanometer,  and  read  the  throw. 

(2)  Charge  again  for  ten  seconds  and  discharge. 

(3)  Repeat  with  increased   time  of  charging  up  to  two  or 
three  minutes,  recording  the  discharge  throw  after  each  charge. 

(4)  Repeat  the  same  program  with  a  paper  condenser  at  C. 

(5)  Tabulate  the  results. 

III.  Residual  Charge.     (1)  With  the  circuit  as  in  Fig.  96, 
charge  a  mica  condenser  for  three  minutes,  then  discharge  and 
note  the  throw.      Bring  the  key  promptly  to  the  insulating 
position,  being  careful  not  to  make  contact  with  the  battery 
terminal.     To  avoid  doing  so,  the  battery  may  be  disconnected. 

(2)  After  successive  periods  of  one  minute  insulation,  dis- 
charge the  condenser  and  note  the  throws. 

(3)  Repeat  the  same  program  with  a  paper  condenser  at  C. 
Include  in  the  report  a  discussion  of  the  information  which 


VIII,  §  158]     WHEATSTONE   BRIDGE   METHOD 


227 


these  experiments  have  yielded  as  to  the  qualities  of  the  con- 
densers used. 

158.  The  Comparison  of  Capacity  with  the  Wheatstone 
Bridge.  FIRST  METHOD.  Two  resistances  and  the  two  con- 
densers to  be  compared  are  ar- 
ranged in  series.  Across  the  alter- 
nate junction  points  a  galvanometer 
and  a  battery  are  connected,  as  shown 
in  Fig.  100.  When  the  three-way 
key  K  is  raised  to  a,  the  condensers 
are  charged,  and  when  pressed  to  b 
they  are  discharged.  The  resist- 
ances RI  and  R2  are  so  adjusted 
that  no  deflection  of  the  galvanom- 
eter occurs  when  the  key  is  ma- 
nipulated. Let  ii  and  i'2  be  the  cur- 
rents through  Rl  and  R2  at  any 
instant,  and  let  V\  and  V2  be  the  corresponding  potential 
differences  across  RI  and  R2  respectively.  Assuming  that 
the  condensers  are  perfect,  the  charges  in  each  are  given  by 
the  equations 


The  condition  for  no  deflection  is   that   the  potential  is  the 
same  at  p  and  p' ;  hence  we  have 


(12) 


Q-2 


If  we  let  dt  represent  a  short  time  interval,  we  may  write 


It 


228     THE  MEASUREMENT  OF  CAPACITY      [VIII,  §  158 
whence 

/1  Q\  H>1        -"2 

(lo)  -^±  =  — -  « 

Combining  equations  (12)  and  (13)  we  have 

(14)  §  =  §' 

It  will  be  observed  that  the  sequence  in  which  the  sides  of 
the  bridge  enter  in  the  formula  differs  from  that  in  the  corre- 
sponding  formula    for    resistance    com- 
parisons. 

SECOND  METHOD.  With  the  con- 
densers and  resistances  arranged  in  a 
closed  circuit  as  before,  the  galvano- 
meter and  battery  are  connected  as  in 
Fig.  101,  with  tap  keys  at  k  and  K. 
When  K  is  closed  the  two  condensers 
are  charged  in  series,  and  the  quantity 
given  to  each  is  the  same.  If  V1  and 
F2  represent  the  charging  potential  differences,  we  may  write 

Qi-ftF,,  Q.=  C2F2, 

whence 

(15)  g  =  £'- 
Moreover,  we  have 


when    no    current    flows   through   the    galvanometer   i1  =  i*2 ; 
hence  we  have 

(16)  ^  =  3' 

Combining  equations  (15)  and  (16)  we  obtain  the  relation 

(17)  £•  =  £. 


VIII,  §  159]     WHEATSTONE  BRIDGE  METHOD 


229 


Absorption  and  leakage  tend  to  give  values  other  than  the 
true  capacity,  and  this  difficulty  is  exaggerated  if  the  time 
of  charge  is  prolonged.  A  uniform  and  quick  tapping  of  the 
keys  will  tend  to  eliminate  this  effect.  For  long  cables  of 
large  capacity  the  time  of  charging  may  be  several  seconds, 
but  for  ordinary  laboratory  condensers  the  keys  should  be 
tapped  in  quick  succession.  An  advantage  of  the  method  is 
that  any  sensitive  galvanom- 
eter, not  necessarily  a  bal- 
listic one,  may  be  used. 

For  cable  testing  the  cir- 
cuit is  shown  in  Fig.  102. 
In  this  case  Oj  is  a  standard 
capacity  and  <72  is  the  cable 
whose  capacity  is  to  be  found. 
The  cable  will  usually  be 
coiled  in  a  tank  and  earth  connections  will  be  made  as  at  E. 
For  rapid  determinations  in  commercial  testing  a  low  frequency 
alternating  current  and  a  telephone  receiver  may  replace  the 
battery  and  galvanometer.  The  time  constants  of  the  arms  Rl 
and  J?2  must  be  small  compared  with  the  period  of  the  alter- 
nating current.  The  vibration  galvanometer  may  also  be  used 
in  place  of  the  telephone  receiver. 

159.  Laboratory  Exercise  XXXII.  To  compare  capacities 
with  the  Wheatstone  bridge.  First  method. 

APPARATUS.  Standard  condenser,  condenser  to  be  meas- 
ured, two  resistance  boxes  of  at  least  2000  ohms  range, 
discharge  key,  several  dry  cells,  and  a  sensitive  galvanom- 
eter. 

PROCEDURE.  (1)  Arrange  the  circuit  as  in  Fig.  100.  Make 
RI  small,  say  10  ohms,  and  make  R^  as  large  as  the  range  of 
the  box  will  permit.  Raise  the  key  to  a,  then  discharge  and 
note  the  throw. 


230     THE  MEASUREMENT  OF  CAPACITY      [VIII,  §  159 

(2)  Proceed  further  as  in  (2),  (3)  and  (4)  of  Laboratory 
Exercise  XXXIII,  §  160. 

The  first  throw  of  the  galvanometer  is  often  followed  by  a  deflection 
in  the  opposite  direction.  This  is  due  to  absorption  and  may  be  disre- 
garded, attention  being  given  to  so  balancing  the  resistances  that  the  first 
throw  is  made  equal  to  zero. 

160.  Laboratory  Exercise  XXXIII.  To  compare  capacities 
with  the  Wheatstone  bridge.  Second  method. 

APPARATUS.  Standard  condenser,  condenser  to  be  tested, 
two  resistance  boxes  of  at  least  2000  ohms  range,  two  tap 
keys,  several  battery  cells,  and  a  sensitive  galvanometer. 

PROCEDURE.  (1)  Arrange  the  circuit  as  in  Fig.  101.  Make 
Rl  small  and  make  R2  as  large  as  the  range  of  the  box  will 
permit.  Tap  k  several  times  to  insure  the  complete  discharge 
of  the  condensers,  then  tap  K  with  k  down,  and  note  the 
deflection. 

(2)  Make  B2  small  and  'Rl  large,  and  repeat  the  procedure. 
Note  the  deflection,  which  will  probably  be  in  the  direction 
opposite  to  the  first  one.     If  it  is  not,  the  range  of  the  resist- 
ance must  be  increased  or  a  standard  condenser  of  different 
value   must  be   selected.     In  case  the  two  deflections  are  in 
opposite  directions,  proceed  to  adjust  R^  and  j£2  until  no  de- 
flection is  observed.     Estimate  and  record  the  least  deflection 
which  could  be  observed  on  the  scale.     Increase  R*  until  this 
least  observable  deflection  appears,  also  decrease  R%  until  the 
deflection   in  the  opposite  direction  is  just  perceptible.     Ke- 
cord  these  values  of  JK2,  also  their  mean,  which  gives  the  most 
probable  value. 

(3)  Eepeat  the  readings  with  different  values  of  Rl  and  R^ 
keeping  them  both  as  large  as  possible  throughout.     The  pre- 
cision of   the  method  is  greater  when  Ci   and  C2  are  nearly 
equal,  and  when  R^  and  J?2  are  high. 

(4)  Calculate  the  value  of  the  unknown  from  equation  (17). 


VIII,  §  161]     WHEATSTONE  BRIDGE  METHOD 


231 


This  method,  being  a  zero  method,  is  superior  to  the  direct  deflection 
method.  High  insulation  of  all  parts  is  necessary.  Good  results  can 
be  obtained  only  when  the  condensers  compared  have  dielectrics  of  the 
same  kind,  for  if  there  is  a  difference  in  the  absorption  of  the  two  con- 
densers it  is  difficult  to  secure  a  balance.  The  effects  of  absorption  may 
be  rendered  less  troublesome  by  making  the  time  of  charge  very  short 
and  by  discharging  promptly. 

161.  Laboratory  Exercise  XXXIV.  To  compare  capacities 
with  the  Wheatstone  bridge  and  vibration  galvanometer. 

APPARATUS.  Condensers  to  be  compared,  Kohlrausch 
bridge,  and  vibration  galvanometer. 

The  Kohlrausch  bridge  was  described  in  §  73.  The  vibra- 
tion galvanometer  is  described  in  §  171. 

PROCEDURE.  (1)  Arrange  the  circuit  as  in  Fig.  103,  with 
the  vibration  galvanometer  and  source  of  alternating  current 
replacing  the  galvanometer  and  bat- 
tery of  the  preceding  method.  The 
resistance  RI  and  J?2  wiH  be  the  two 
segments  of  the  long  wire  of  the 
Kohlrausch  bridge.  One  terminal  of 
the  vibration  galvanometer  is  con- 
nected to  the  movable  contact  of  the 
bridge,  and  the  other  is  connected  at 
the  junction  of  the  two  condensers. 

(2)  Adjust   Rl  and  R2  until   the 
band  of  light  is  sharply  defined  and 
of  minimum   width.      Vary  the    re- 
sistances slightly  until  the  least  observable  broadening  appears, 
and  estimate  the  precision  of  the  settings. 

(3)  Calculate  the  results  by  means  of  equation  (17). 

Uncompensated  inductance  in  the  bridge  arms  will  prevent  a  balance, 
and  the  wire  turns  of  the  Kohlrausch  bridge  are  not  non-inductive. 
However,  as  the  lengths  of  the  segments  change,  the  resistances  change 
in  equal  ratio,  and  the  time  constants  remain  essentially  equal.  Hence, 
sharp  settings  are  easily  made. 


FIQ.  103. 


232     THE  MEASUREMENT  OF  CAPACITY      [VIII,  §  162 

162.  The  Comparison  of  Capacities  by  the  Method  of 
Mixtures.  This  method  consists  essentially  in  (a)  charging 
two  condensers  in  series  by  a  potentiometer  method,  (b)  so 
connecting  them  that  the  charges  will  mix,  tending  to  neutral- 
ize one  another,  and  (c)  testing  for  the  resultant  or  outstand- 
ing charge,  if  there  is  any. 

In  Fig.  104,  Ci  is  the  condenser  to  be  tested,  C2  is  the  con- 
denser of  known  capacity,  R^  and  R2  are  resistance  boxes,  B  is 

a  battery  of  several 
cells,  g  is  a  ballistic 
or  other  sensitive  gal- 
vanometer, and  S  is  a 
highly  insulated  dou- 
ble-pole, double-throw 
switch  for  the  purpose 
FlG>  104-  of  making  the  connec- 

tions in  the  proper  order.  The  switch  is  so  arranged  that 
when  the  handle  is  thrown  to  the  right  the  contact  at  5  is 
made  an  instant  before  that  at  6.  The  wires  that  cross  at 
D  are  connected.  The  point  D  is  sometimes  connected  to 
ground. 

Whatever  the  position  of  the  switch,  a  current  i  is  always 
flowing  through  both  resistances,  and  if  the  switch  is  thrown 
so  as  to  connect  1  with  3  and  2  with  4,  both  condensers  will 
be  charged.  The  condenser  Ci  is  charged  with  the  potential 
difference  Fi  that  exists  across  the  terminals  of  R19  and  C2  is 
charged  with  the  potential  difference  V2  that  exists  across  the 
terminals  of  R2.  If  the  switch  is  thrown  to  the  right,  as  soon 
as  3  is  connected  to  5  the  charges  in  the  two  condensers  will 
mix,  for  the  plates  a  and  b  are  already  connected  and  the 
switch  now  makes  connection  between  c  and  d. 

If  the  charges  in  Ci  and  C2  are  equal,  the  positive  charge 
from  c  will  exactly  neutralize  the  negative  charge  from  d,  and 
the  negative  charge  from  a  will  neutralize  the  positive  charge 


VIII,  §  162]          METHOD   OF  MIXTURES  233 

from  b.  For  this  condition  no  charge  will  remain  in  the 
condensers. 

But  if  the  charges  in  d  and  C2  are  not  equal,  both  con- 
densers will  still  be  charged  to  some  extent.  An  instant  after 
the  switch  makes  contact  between  3  and  5  it  connects  4  and 
6,  and  when  this  occurs  the  condensers  are  both  discharged 
through  the  galvanometer. 

After  the  switch  has  been  thrown  to  the  left,  if  no  deflection 
of  the  galvanometer  occurs  on  rocking  the  switch  to  the  right, 
we  may  conclude  that  when  the  switch  was  thrown  to  the  left, 
the  condensers  were  charged  with  equal  quantities.  In  this 
case  we  have 


Moreover,  we  know  that  Vl  =  iJRl}  and  V2  =  iR^  where  i  is 
the  steady  current  flowing  from  the  battery.  We  may  then 

write 

dEji  =  C2R2i, 

whence 

(18)  C,=fQ. 

Hi 

If  the  capacity  to  be  determined  is  that  of  a  cable,  the  nec- 
essary time  for  charging  may  be  a  minute  or  even  longer. 
For  ordinary  condensers  a  second  or  two  is  more  than  enough. 

The  particular  advantages  of  the  method  are  that  it  is  less 
affected  by  absorption  than  the  methods  of  the  preceding  arti- 
cles, and  that  it  is  applicable  over  a  wide  range  of  values. 

If  absorption  is  troublesome,  interchange  the  battery  termi- 
nals and  again  adjust  for  a  zero  deflection.  Use  the  mean 
of  the  two  resistance  settings.  This  effect  will  be  reduced  by 
a  rapid  rocking  of  the  switch. 

The  method  of  mixtures  is  probably  the  most  widely  used 
of  all  comparison  methods  for  the  calibration  of  standards, 
precision  comparisons,  and  practical  cable  testing. 


234     THE  MEASUREMENT  OF  CAPACITY     [VIII,  §  163 

163.  Laboratory  Exercise  XXXV.  To  compare  capacities 
by  the  method  of  mixtures. 

APPARATUS.  Two  resistance  boxes,  standard  condenser  and 
condenser  to  be  measured,  several  dry  cells,  sensitive  galva- 
nometer, and  special  switch. 

PROCEDURE.  (1)  Arrange  the  circuit  as  in  Fig.  104,  being 
careful  that  all  wires  are  air  lines,  not  resting  on  the  apparatus 
nor  touching  one  another. 

(2)  In  seeking  the  correct  values  of  Rl  and  R2  for  securing 
equal   charges   in   the  condensers,  operate   systematically   as 
follows.     Make  Rl  small  and  R2  as  large  as  possible.     Throw 
the  switch  to  the  left  and  then  to  the  right  and  note  the 
direction  of  the  galvanometer  deflection.     Then  make  R«  small 
and  R±  as  large  as  possible  and  repeat  the  procedure.     The 
deflections  will  probably  be  in  opposite  directions.     If  not,  the 
capacities  are  probably  too  widely  divergent  for  the  resistance 
ranges  used.     This  range  must  then  be  extended,  or  a  standard 
capacity  nearer  to  the  unknown  must  be  chosen. 

If  the  deflections  are  opposite  in  direction  it  is  obvious  that 
the  values  of  the  resistances  for  zero  deflections  lie  between 
the  extreme  limits,  and  these  values  are  readily  found. 

Since  the  charge  deflecting  the  galvanometer  is  a  differential 
one  it  is  obvious  that  the  sensibility  of  the  method  increases 
with  the  E.  M.  F.  of  the  battery  used.  The  values  of  Rl  and 
R2  should  not  be  less  than  one  or  two  thousand  ohms. 

(3)  Estimate  and  record  the  least  observable  deflection  that 
can  be  read  on  the  galvanometer  scale.     When  values  of  the 
resistances  are  found  for  the  condition  of  zero  deflection,  keep 
RI  constant  and  find  what  changes  in  R2  will  give  the  least 
perceptible    deflection    to    the    right    and    left,    respectively. 
Substitute  the  mean  of  these  values  of  R2  in  equation  (18). 

(4)  Repeat  with  several  different  sets  of  values  of  RI  and  R%. 
Tabulate  all  data  and  results,  the  probable  precision  of  the 
readings,  and  the  percentage  accuracy  of  the  results. 


VIII,  §  164]  HIGH    RESISTANCE  235 

164.  The  Discharge  of  a  Condenser  through  a  High 
Resistance.  If  a  condenser  of  capacity  C  is  charged  with  an 
initial  potential  difference  F0,  and  then  is  allowed  to  discharge 
through  a  very  high  resistance  R,  the  charge  will  gradually 
disappear  and  the  potential  difference  between  the  plates  will 
slowly  sink  to  zero. 

Let  V  be  the  instantaneous  value  of  the  potential  difference 
at  some  time  t  seconds  after  the  initial  charge.  The  quantity 
in  the  condenser  at  this  instant  is  given  by  Q  =  OF,  and  the 
rate  at  which  the  charge  is  decreasing  is  given  by  the  equation 


(19)  _       =  _ 

dt  dt 

The  instantaneous  value  of  the  decreasing   current  may  be 
written  in  the  form 


hence,  combining  (19)  and  (20),  we  have 


Separating  the  variables,  we  have 
(22) 


OR 


Integrating  both  sides  of  this  equation,  and  remembering  that 
when  t  =  0,  F=  F0  ,  we  find 


Solving  this  equation  for  R,  we  find 
(23)  R= 1 


236      THE   MEASUREMENT   OF  CAPACITY     [VIII,  §  164 

Since  galvanometer  throws  are  proportional  to  charges,  and 
since  these  are  proportional  to  potential  differences,  the  ratio 
d0/d  may  replace  VJV  in  equation  (23).  Again,  the  potential 
differences  V0  and  V  may  be  measured  directly  by  means  of  an 
electrostatic  voltmeter  or  an  electrometer,  and  the  ratio  F0/  V 
may  be  calculated  from  these  measurements. 

This  method  is  used  where  the  resistance  to  be  measured  is 
so  high  that  the  direct  deflection  method  of  §  69  cannot  be 
used.  At  best,  the  method  is  subject  to  many  errors,  and  the 
results  of  different  tests  on  the  same  sample  are  frequently 
not  in  good  agreement. 

It  is  known  that  for  most  insulators  the  leakage  current 
does  not  follow  Ohm's  law.  Moreover,  this  variation  is  a 
function  of  the  impressed  voltage.  In  a  cable,  the  resistance 
of  the  dielectric  substance  changes  with  the  temperature, 
decreasing  as  the  temperature  rises. 

The  impressed  potential  difference  seems  to  produce  certain 
molecular  changes  in  the  material  of  the  nature  of  polarization, 
so  that  the  resistance  of  the  insulation  increases  steadily  after 
the  potential  difference  is  applied.  This  phenomenon  is  called 
electrification.  It  is  customary  to  specify  a  certain  time, 
usually  one  minute,  as  the  time  of  electrification. 

Again,  if  the  cable  is  immersed  in  water,  the  time  for  which 
it  has  been  immersed  must  be  specified.  To  say  that  a  cable 
has  an  insulation  resistance  of  100  megohms  per  mile  is 
only  significant  when  the  conditions  of  the  test  are  carefully 
stated. 

The  value  of  an  insulation  resistance  must  then  be  accom- 
panied by  a  statement  of 

(a)  the  temperature, 

(6)    the  time  of  electrification, 

(c)  the  time  of  immersion, 

(d)  the  impressed  potential  difference. 
The  dielectric  strength  also  is  usually  specified. 


VIII,  §  165] 


HIGH  RESISTANCE 


237 


165.  Laboratory  Exercise  XXXVI.  To  measure  a  high  re- 
sistance by  the  loss  of  charge  method. 

APPARATUS.  Sample  of  cable  in  tank  of  water,  ballistic 
galvanometer,  high  potential  battery,  discharge  key,  and 
standard  condenser. 

PROCEDURE.  (1)  Arrange  the  circuit  as  in  Fig.  105.  The 
sample  cable  of  capacity  C  is  represented  by  AB. 

If  the  cable  is  lead  covered  the  contact  c  may  be  made 
directly  to  the  sheath,  B 
being  connected  to  the  core. 
Otherwise,  a  tank  of  water 
with  connection  shown  at 
c  is  required.  Using  a 
suitable  voltage,  calibrate 
the  galvanometer  with  the 

standard  condenser,  then  charge  and  discharge  C  and  compute 
its  capacity  from  equation  (9). 

(2)  Press  K  to  a  for  one  minute,  then  quickly  press  to  6, 
and  read  the  throw  d0  on  the  galvanometer.     Keep  K  on  b  for 
a  minute  or  two  until  C  is  fully  discharged,  then  charge  for 
one  minute  and  set  the  key  on  the  insulating  position  for  a 
few  seconds.     Then   press  K  to   6,  and  read   the   throw  d. 
Select  the  time  for  insulating  such  that  d  is  approximately 
half  dv 

(3)  Calculate   the   value   of    R  from  equation   (23).     The 
length  of  the  sample  will  be  measured  and  the  result  reduced 
to  megohms  per  mile. 

In  case  only  a  short  sample  is  available,  the  method  should  be  modified 
as  follows.  Across  the  terminals  of  the  sample  AB  connect  in  parallel  a 
condenser  of  known  capacity  C1.  Assume  the  capacity  of  the  small  piece 
of  cable  to  be  negligible  and  use  <7  as  the  capacity  of  the  system.  It  will 
be  necessary  first  to  determine  the  insulation  resistance  of  (7,  which  may 
be  called  R'.  Then,  by  the  method  described  above,  find  R",  the  joint 
resistance  of  C  and  C"  in  parallel.  From  the  formula  for  resistances  in 
parallel,  calculate  the  insulation  resistance  B  of  C  alone. 


CHAPTER  IX 


THE   MEASUREMENT   OF    SELF   AND    MUTUAL 
INDUCTANCE 

PART  I.     SELF-INDUCTANCE 

166.   General    Methods.     Three    methods    used    for     the 
measurement  of  self-inductance  are : 

I.  Absolute  methods,  in  which  the  value  is  found  independ- 
ently of  any  previously  measured  inductance. 

II.  Capacity-comparison   methods,  in  which   the   unknown 
inductance  is  measured  in  terms  of  a  standard  capacity. 

.  III.  Inductance-comparison  methods,  in  which  the  unknown 
inductance  is  measured  by  direct  comparison  with  a  standard. 

Two  representative  methods  under  I 
and  II  will  be  described,  the  formulas 
being  given  without  proof. 

In  the  Maxwell- Rayleigh  method, 
the  inductance  is  measured  in  terms  of 
a  resistance  and  a  time.  The  induc- 
tance L  which  is  to  be  measured  is 
put  in  one  arm  of  a  Wheatstone  bridge 
circuit  (Fig.  106),  the  other  arms  being 
non-inductive.  The  bridge  is  balanced 
for  a  steady  current  in  the  usual  way. 
The  key  K  is  first  closed,  then  Kf  is 

tapped,  and  the  resistances  are  adjusted  until  there  is  no 
deflection.  When  a  balance  has  been  obtained,  K'  is  closed, 
and  the  deflection  dj,  which  occurs  when  Kis  tapped,  is  read. 

238 


Fia.  100. 


IX,  §  166] 


MAXWELL'S  METHOD 


239 


The  steady  current  balance  is  then  disturbed  by  altering  JR^ 
by  some  small  amount  r,  and  the  steady  deflection  d2,  which 
occurs  when  both  keys  are  closed,  is  read.  If  the  steady 
currents  through  ^x  and  Rs  are  denoted  by  it  and  i3,  respect- 
ively, if  T  is  the  period  of  the  moving  needle  of  the  galvanom- 
eter, and  if  it  is  assumed  that  there  is  no  damping,  the  value 
of  L  is  given  by  the  formula 


(1) 


L=rT 


The  ratio  of  the  currents  may  be  computed  from  the  known 
resistances  in  the  circuit  and  the  E.  M.  F.  of  the  battery  used. 

The  value  of  L  is  thus  seen  to  be  given  in  terms  of  pure 
numbers  together  with  a  resistance  and  a  time.  Since  resistance 
has  dimensions  \_LT~l~],  the  dimen- 
sional formula  for  inductance  be- 
comes [7v],  whence  the  absolute 
unit  is  called  the  centimeter. 
The  practical  unit,  which  is  109 
centimeters,  is  called  the  henry. 
Formerly  it  was  called  the  secohm 
because  it  could  be  expressed  in 
terms  of  time  and  resistance,  as  in 
equation  (1). 

In  Maxwell's  method  of  deter- 
mining an  inductance  in  terms  of 
a  capacity,  a  Wheatstone  bridge 
circuit  is  arranged  with  three  non-inductive  arms,  and  with  the 
inductance  L  in  the  fourth  arm,  as  shown  in  Fig.  107.  The 
bridge  is  balanced  in  the  usual  way  for  steady  currents,  press- 
ing Kf  only  after  K  has  been  closed.  The  presence  of  the 
condenser  across  the  arm  jfi^  will  have  no  effect  on  this  balance. 
If  K  is  now  tapped  after  K1  has  been  closed,  there  will  be  a 


FIG.  107. 


240      MEASUREMENT   OF  SELF-INDUCTION    [IX,  §  166 


deflection  on  the  galvanometer,  and  this  may  be  made  zero  by 
a  new  adjustment  of  Hly  which  changes  the  charging  potential 
across  C.  When  the  galvanometer  does  not  deflect  when  either 
key  is  tapped  first,  it  can  be  shown  that  L  is  given  by  the 
formula 

(2)  L  =  CRJtt. 

This  formula  is  very  simple,  but  the  double  adjustment  is 
extremely  tedious,  inasmuch  as  the  change  in  EI  upsets  the 
previous  balance  obtained  with  a  steady  current.  With  a 
suitable  resistance  box  provided  with  traveling  plugs,  the  con- 
tact a  may  be  moved  to  any  position  on  R^  thus  varying  the 
charging  potential  without  annulling  the  bridge  balance. 

Another  way  of  avoiding  the  necessity  for  the  double  adjust- 
ment is  to  use  at  C  a  condenser  whose  capacity  may  be  varied 
either  by  steps  or  continuously.  This  form  of  condenser  is  not 
usually  available.  The  best  modification  of  the  method  is  one 
that  will  be  described  fully  in  §§  169,  170. 

167.   Inductance  in  Terms  of  a  Capacity.     Consider  a  cir- 
cuit arranged  as  in  Fig.  108,  in  which  aa  is  an  electromagnet 
of  large  inductance  in  series  with  a  source 
of  current  D.     In  parallel  with  the  induc- 
tance is  a  lamp  p. 

Suppose  the  current  to  be  adjusted  so 
that  when  K  is  closed  the  lamp  glows  a 
dull  red.  Then  on  opening  the  switch  K 
the  brightness  of  the  lamp  filament  is  first 
suddenly  and  very  greatly  increased  for 
an  instant,  and  then  quickly  diminished. 
The  sudden  rise  of  current  through  the 
lamp  is  due  to  the  energy  stored  in  the  magnetic  field,  which  is 
returned  to  the  circuit  when  K  is  opened.  The  inductance 
develops  a  potential  difference  at  the  terminals  which  is  greater 
than  that  originally  impressed,  as  shown  in  §  136. 


IX,  §  167] 


MAXWELL'S   METHOD 


241 


If  a  galvanometer  or  other  measuring  instrument  could  be 
inserted  at  p,  the  flow  of  charge  could  be  measured  and  the 
value  of  the  inductance  could  be  calculated.  However,  any 
such  instrument  at  p  would  be  given  a  steady  deflection  when 
K  was  closed  ;  hence  it  would  not  be  in  a  condition  to  receive 
the  charge  when  K  was  opened. 

It  is  necessary,  therefore,  to  devise  some  method  whereby 
the  galvanometer  can  be  in  its  zero  position  until  the  instant 
that  K  is  opened.  This  is  ac- 
complished in  a  simple  manner 
by  making  the  inductance  one  arm 
of  a  Wheatstone  bridge  circuit  (Fig. 
109) .  With  the  bridge  balanced  for 
a  steady  current  the  galvanometer 
terminals  ab  will  be  at  the  same 
potential,  hence  there  will  be  no 
deflection.  WThen  K  is  opened  the 
energy  stored  in  the  magnetic  field 
about  R  is  given  back  to  the  circuit, 
and  the  quantity  of  electricity 
thereby  induced  is  discharged,  partly  through  the  galvanometer 
and  partly  through  the  rest  of  the  circuit. 

The  value  for  L  may  be  found  as  follows.  With  the 
bridge  balanced  for  steady  current,  the  galvanometer  shows 
no  deflection  when  K  is  kept  closed.  Assume  a  current 
of  strength  /  flowing  in  the  arm  R3.  When  K  is  opened 
this  current  drops  to  zero  and  the  magnetic  field  about  L 
collapses.  The  total  change  N  in  the  number  of  linkings 
due  to  a  current  change  from  /  to  zero  is,  by  equation 
(22),  §  133, 


FIG.  109. 


(3) 


N=LL 


Figure  110  represents  exactly  the  same  circuit  as  that  shown 
in  Fig.  109.     A  study  of  this  diagram  will  show  that  the  total 


242      MEASUREMENT  OF  SELF-INDUCTION    [IX,  §  167 

quantity  of  electricity  discharged  through  the  entire  circuit  is, 
by  equation  (76),  §  142, 


where  R  is  the  equivalent  resistance  around  aAbE. 

Suppose  that  the  arm  Rz  is  now  removed  from  the  bridge 
and  replaced  by  a'E'  (Fig.  110),  in  which  the  resistance  S  +  r 
is  exactly  equal  to  R3)  and  around  some  part  of  which,  $,  is 

C 


shunted  a  condenser  of  capacity  C.  When  K  is  kept  closed 
the  same  current  as  before  flows  through  r  and  S,  since  the 
bridge  balance  has  not  been  disturbed.  The  charging  poten- 
tial across  C  is  then  IS,  and  the  quantity  in  the  condenser  is 
given  by  the  equation 

(5)  Q'2=CV=CIS. 

If  K  is  opened  this  quantity  is  discharged,  part  passing 
through  S  and  part  through  all  the  rest  of  the  circuit,  or  R,  -f-  r. 
That  part  which  passes  through  R  is  then  given  by  the 
equation 

(6)  Q*=CIS(R     S -. 

When  the  quantities  Ql  and  Q2  are  discharged  through  the  cir- 
cuit, the  corresponding  galvanometer  deflections  may  be  called 


IX,  §  168]  MAXWELL'S  METHOD  243 

dl  and  d2  respectively.  However,  these  deflections  are  not 
caused  by  the  entire  charge,  because  the  galvanometer  is  in 
parallel  with  other  resistances,  as  shown  in  Fig.  110.  Since 
these  resistances  are  constant,  the  fraction  of  the  total  charge 
is  the  same  in  each  case  and  the  charges  Qi  and  Q.2  are  propor- 
tional, respectively,  to  the  galvanometer  throws.  Dividing 
(4)  by  (6),  we  have 


Q2         CIS* 
fi  +  r  + 
and  since 

R3  +  R  =  It 

we  may  write  (7)  in  the  form 

L       d 


or 

(8)  L 

z 

The  shunted  condenser  may  be  put  in  series  with  the  in- 
ductance L,  and  a  balance  secured  by  varying  the  charging 
potential  across  the  condenser  by  means  of  the  traveling  con- 
tact at  c  (Fig.  109).  This  procedure  has  the  advantage  of 
being  a  zero  method,  and  equation  (8)  then  takes  the  form 

(9)  L  =  OS2. 

168.  Laboratory  Exercise  XXXVII.  To  determine  an  induc- 
tance in  terms  of  a  capacity. 

APPARATUS.  Box  bridge  and  portable  galvanometer,  bal- 
listic galvanometer,  a  few  dry  cells,  special  double-break  key, 
standard  condenser,  resistance  box  with  side  plugs,  and  the 
inductance  to  be  measured. 

PROCEDURE.  (1)  Arrange  the  circuit  as  in  Fig.  109.  Con- 
nect the  inductance  across  the  line  terminals  of  the  bridge  box. 


244      MEASUREMENT  OF  SELF-INDUCTION    [IX,  §  168 

With  the  battery  and  portable  galvanometer  properly  con- 
nected, balance  the  bridge  to  the  nearest  ohm,  using  a  ratio 
of  100  to  100.  Replace  the  portable  galvanometer  by  the 
ballistic  one,  and  connect  the  double-break  key  K1  so  that  a 
short-circuit  across  the  galvanometer  terminals  is  broken  an 
instant  before  the  battery  circuit  is  opened.  With  the  bridge 
keys  closed  and  the  galvanometer  quiet  at  zero,  quickly  press 
K'  and  read  the  deflection  d^. 

(2)  Replace  the   inductance  by  a  non-inductive  resistance 
of  equal  value,  and  shunt  around  all  or  part  of  it  the  capacity 
C.     Let  S  be  the  value  of   this  shunting   resistance.     Again 
close  the  bridge  keys  and  quickly  press  K1,  reading  the  throw 
d2,  which  should   be  in  the   opposite  direction  to  d},  and   of 
approximately  the  same  magnitude. 

In  order  to  insure  a  constant  torsion  of  the  suspension  it 
is  well  to  reverse  the  galvanometer  terminals  before  taking 
d2,  so  that  both  throws  are  read  on  the  same  side  of  the  zero. 
If  di  is  much  too  large,  inserting  a  non-inductive  resistance 
in  series  with  L  will  reduce  it,  in  which  case  the  bridge  must 
be  balanced  again. 

If  d2  is  too  small,  it  may  be  increased  by  increasing  the 
value  of  either  S  or  C.  The  total  resistance  of  that  arm  of 
the  bridge  must,  however,  be  kept  the  same  when  both  di  and 
d2  are  observed. 

(3)  After  a  few  trials  to  find  the  best  working  conditions, 
repeat  both  dl  and  d2  several  times  and  take  the  respective 
means.     Calculate  L  from  equation  (8).     Express   the  value 
in  henrys  and  also  in  millihenrys. 

Read  again  §  167,  and  note  why  the  Wheatstone  bridge  circuit  is  used 
here.  It  is  a  difficult  matter  to  maintain  an  accurate  bridge  balance,  thus 
keeping  the  galvanometer  precisely  on  zero  without  drifting.  By  the  use 
of  the  double-break  key,  however,  this  precise  balance  is  not  necessary. 
The  short-circuit  keeps  the  potential  at  a  and  b  the  same,  and  the  double- 
break  key  is  so  adjusted  that  the  short-circuit  is  broken  a  very  short  time 
before  the  battery  circuit  is  opened.  This  time  interval  should  be  made 


IX,  §  169] 


ANDERSON'S  METHOD 


245 


so  short  that  the  galvanometer  is  not  appreciably  deflected  meanwhile  by 
any  slight  lack  of  balance  which  may  exist.  Obviously,  after  the  battery 
circuit  is  broken,  this  lack  of  balance  can  have  no  further  influence  upon 
the  galvanometer,  which  is  then  deflected  solely  by  the  quantity  dis- 
charged through  its  coils. 

169.  Inductance  in  Terms  of  a  Capacity.  A  Modification 
of  Anderson's  Method.  In  §  166  it  was  pointed  out  that  the 
method  due  to  Maxwell  is  tedious  because  of  the  annulling  of 
the  bridge  balance  when  the  charging  potential  across  the  con- 
denser is  changed.  The  following  modification  of  Anderson's 
method  is  free  from  this  objection. 

Arrange  the  circuit  as  in  Fig.  111.  The  inductance  L  is 
made  one  arm  of  the  Wheatstone  bridge  and  its  resistance  is 
found  in  the  usual  way, 
the  bridge  being  balanced 
with  the  greatest  care. 
The  presence  of  S  and 
C  will  not  in  any  way  d 
affect  the  bridge  balance 
for  steady  current. 

If,  after  the  bridge  is 
balanced,  the   key   K  is 

closed  and   k  is  tapped,  FlG  m 

there  will  be  no  deflection 

of  the  galvanometer.  However,  if  k  is  closed  and  K  is  tapped, 
there  will  be  a  deflection.  This  deflection  can  be  made  to 
vanish  by  properly  adjusting  S  and  C.  When  these  have  been 
set  so  that  the  galvanometer  shows  no  deflection  for  either 
order  of  tapping  the  keys,  L  may  be  calculated  from  equation 
(18),  below. 

The  condition  for  the  steady  current  balance  is 

(10)  PN=RM. 

The  currents  through  ae,  ab,  be,  and  ce  may  be  represented 


246      MEASUREMENT  OF  SELF-INDUCTION    [IX,  §  169 

respectively   by  J1?  /2,  /2>  &n(l  /a-      The   potential   difference 
across  be  equals  MI% ;  hence,  for  the  balanced  condition,  we  have 

(11)  PI3  =  MI2. 

Also,  for  a  balanced 
condition  with  variable 
a  current  the  potential  dif- 
ference across  ae,  which 
is  equal  to  that  across 
ab,  is  given  by  the  Helm- 
holtz  equation, 

(12)  RIi  =  N. 


K 


FIG.  Ill  (repeated). 

The  potential  difference   V  across  the  condenser  terminals 
de  is  given  by  the  equation 

(13)  V  =  S(I2  4  Is)  +  P/s  =  SI2  4  SI i  +  P/S. 
The  charge  in  the  condenser  is 

(14)  Q  =  CV=  C\_SI2  4  SI3  4  P/s]. 


The  instantaneous  current  through  the  condenser  is  given 
by  the  equation 


whence 

(15)  , 

From  (11)  we  have 


dt  dt ' 

whence  (15)  may  be  written  in  the  form 

SMdL 


(16) 


^  + 


M 
Pdt  dt 


IX,  §  170]  ANDERSON'S  METHOD  247 

If  both  sides  of  (16)  are  multiplied  by  It,  the   right-hand 
members  of  (12)  and  (16)  may  be  equated,  and  we  have 


(17)  N 

CLt 

But,  by  (10),  N=MR/P;    hence,  equating   coefficients  of 
dI2/dt,  we  have 


or 

(18)  L  =  C[RM  +  RS  +  N8]. 

If  S  is  made  zero  the  equation  reduces  to  L  =  CRM,  which  is 
the  same  as  equation  (2). 

Referring  again  to  Fig.  Ill,  and  assuming  that  the  polarity 
of  the  battery  is  as  marked,  the  steady  currents  through  the 
bridge  will  be  in  the  directions  of  arrows  1,  2,  3,  and  4.  When 
the  key  K  is  opened,  the  effect  of  the  self-inductance  L  is  to 
raise  the  potential  at  6.  However,  at  the  same  time  the  con- 
denser is  discharging  in  the  direction  of  arrow  5,  which  tends 
to  raise  the  potential  at  e.  Increasing  S  increases  the  charg- 
ing potential  across  (7,  but  at  the  same  time  decreases  the 
current  through  L.  Decreasing  the  resistance  of  S  reverses 
these  conditions.  If  S  is  small  compared  to  P  and  M,  and 
if  the  condition  L  >  CRM  is  fulfilled,  then  the  potentials  at 
e  and  6  may  be  made  equal  and  the  galvanometer  does  not 
deflect. 

170.  Laboratory  Exercise  XXXVIII.  To  determine  an  induc- 
tance by  a  modification  of  Anderson's  method. 

APPARATUS.  Four  resistance  boxes,  standard  condenser, 
several  dry  cells,  two  tap  keys,  sensitive  galvanometer,  and 
inductance  to  be  measured. 

PROCEDURE.  (1)  Arrange  the  circuit  as  in  Fig.  111.  With 
S  equal  to  zero  balance  the  bridge  in  the  usual  way,  always 
tapping  key  k  after  K  has  been  closed. 


248      MEASUREMENT  OF  SELF-INDUCTION    [IX,  §  170 

(2)  Reverse  the  order  of  tapping  the  keys  and  adjust  S  and 
C  until  no  deflection  is  observed.     Some  time  will  be  required 
to  find  the  correct  value  of  S.     The  bridge  balance  may  not 
be  preserved  due  to  temperature  changes ;  hence,  it  is  neces- 
sary to  readjust  the   steady  current  balance.     Several  differ- 
ent values  of  the  capacity  should  be  used  as  well  as  different 
values  of  the  ratio  arms  of  the  bridge. 

(3)  Calculate  the  value  of  L  from  equation  (18)  and  indi- 
cate the  probable  precision  of  the  various  quantities,  as  well 
as  of  the  value  of  L.     The   data  and  the  calculated  results 
should  be  tabulated. 

It  can  be  shown  that  the  method  is  most  sensitive  when  P  and  M  are 
large  and  when  S  is  small.  If  equation  (18)  is  solved  for  S,  we  may 
write 


whence  it  is  seen  that  L/C  must  be  greater  than  EM  in  order  to  make  S 
a  positive  quantity.  Hence,  the  values  of  the  various  quantities  must  be 
chosen  subject  to  the  condition  that  L  exceeds  CEM. 

The  vibration  galvanometer,  §  171,  and  a  source  of  low  frequency 
alternating  current,  may  replace  the  galvanometer  and  the  battery  with 
a  considerable  gain  in  precision. 

171.  The  Vibration  Galvanometer.  In  the  ordinary  mov- 
ing coil  galvanometer,  the  deflection  is  caused  by  the  reaction 
of  the  magnetic  field  due  to  a  steady  current  in  the  coil  with 
the  permanent  field  of  the  fixed  magnet,  and  the  deflection  is 
proportional  to  the  strength  of  the  current. 

If  an  alternating  current  is  sent  through  the  movable  coil, 
the  torque  will  reverse  with  each  change  in  direction  of  the 
current  and  the  resultant  motion  is  essentially  zero  because 
of  the  large  inertia  of  the  suspended  system. 

In  case  the  suspended  coil  is  made  very  narrow  and  ex- 
tremely light,  with  both  the  inertia  and  the  damping  small,  it 
will  follow  the  reversed  torque  due  to  the  alternating  current, 


IX,  §  171]    THE  VIBRATION  GALVANOMETER  249 

and  a  beam  of  light  from  a  linear  source  reflected  from  a 
small  mirror  attached  to  the  coil  will  be  spread  out  into  a 
light  band  on  the  scale. 

This  is  the  arrangement  in  the  so-called  vibration  galva- 
nometer. Its  construction  is  quite  like  that  of  the  ordinary 
d'Arsonval  type,  but  differs  from  it  in  having  a  very  narrow 
and  light  coil  of  fine  wire  hung  from  a  bifilar  suspension, 
which  is  capable  of  adjustment  in  both  tension  and 
length.  These  adjustments  can  vary  the  natural 
frequency  of  the  suspended  system  through  wide 
limits,  usually  from  50  to  1000  per  second,  and  its 
sensibility  will  be  high  when  the  natural  frequency 
is  tuned  to  agree  with  that  of  the  alternating  cur- 
rent through  its  coil.  In  this  condition  of  resonance 
a  feeble  current  will  cause  a  broad  band  of  light  on 
the  scale,  and  for  frequencies  differing  but  little 
from  this  value  the  sensibility  falls  off  rapidly. 

The  vibration  galvanometer  is  valuable  also  for 
zero  methods  with  alternating  currents,  since  har- 
monics in  the  current  wave  have  little  effect  so  that 
the  conditions  of  a  pure  sine  wave  may  be  assumed. 

Figure  112  shows  the  arrangement  of  the  parts. 
The  current  is  conducted  to  and  from  the  coil  cc' 
by  the  bifilar  suspension  S  and  S'.     The  attached 
concave  mirror  M  forms  an  image  of  a  linear  source    '  Fio7ii2. 
of  light  on  the  scale,  and  the  width  of  the  bright 
band  on  the  scale  is  a  measure  of  the  amplitude  of  vibration 
of  the  coil. 

The  method  of  tuning  the  vibration  galvanometer  is  similar 
to  that  of  tuning  a  violin.  The  supporting  head  A  can  be 
raised  or  lowered  by  means  of  an  adjusting  screw,  thus  alter- 
ing the  tension  of  the  suspension.  A  second  adjusting  screw 
slides  the  ivory  fret  B  up  or  down  along  the  wires,  thus  alter- 
ing the  effective  length  of  the  suspension.  In  this  manner, 


250       MEASUREMENT  OF  SELF-INDUCTION    [IX,  §  172 


the  natural  frequency  of  the  system  is  brought  into  resonance 
with  the  impressed  alternating  current.  The  sensibility  of 
the  vibration  galvanometer  is  specified  in  terms  of  the  current 
strength  which  causes  one  millimeter  of  broadening  of  the 
image  of  a  linear  bright  source,  with  the  scale  at  a  distance  of 
one  meter  from  the  mirror.  A  current  strength  of  10~6  ampere 
can  be  measured,  and  a  current  of  10~7  ampere  can  be  detected. 

172.   Comparison  of    Self -Inductance.     Vibration  Galva- 
nometer Method.     The  variable  standard  of  self-inductance 

consists  of  two 
coils  wound  on 
spherical  shells 
and  connected  in 
series,  the  same 
current  passing 
through  both 
coils.  A  common 
form  is  shown  in 
Fig.  113.  One  of 
these  coils  lies 
within  the  other 
with  the  least  pos- 
sible clearance, 
and  is  capable  of 
rotation  about  a 
vertical  axis 
through  an  angle 
of  180  degrees, 
its  position  being 
indicated  by  a  pointer  which  plays  over  a  horizontal  scale 
on  top  of  the  frame. 

If  the  movable  coil  is  so  placed  that  its  magnetic  field  re- 
inforces the  field  of  the  fixed  coil,  the  number  of  linkings,  and 


FIG.  113. 


IX,   §  172]    THE  VIBRATION   GALVANOMETER 


251 


hence  the  self-inductance  of  the  system,  is  a  maximum.  If, 
however,  the  movable  coil  is  rotated  180  degrees  from  this 
position,  its  field  will  nearly  or  quite  annul  that  of  the  fixed 
coil  and  the  number  of  linkings  and  hence  the  inductance  will 
be  correspondingly  reduced.  For  intermediate  positions  the 
pair  of  coils  will  have  inductance  values  varying  from  nearly 
zero  to  the  maximum,  and  these  values  can  be  read  from  the 
graduated  scale.  In  the  instrument  here  used  this  range  is 
from  0.005  to  0.040  henry. 

In  the  circuit  shown  in  Fig.  114,  L±  is  the  standard  variable 
inductance  of  resistance  Rlt  L2  is  the  inductance  to  be  meas- 


FIG.  114. 

ured  of  resistance  B2>  and  R3  and  R±  are  non-inductive  resist- 
ances. A  double-pole  double-throw  switch  W  is  arranged  so 
as  to  connect  either  a  dry  cell  B  or  a  source  of  alternating  cur- 
rent A.  C.  to  the  bridge  at  a  and  b. 

Another  double-pole  double-throw  switch  U  connects  the 
other  terminals  of  the  bridge  cd  to  an  ordinary  galvanometer 
P,  or  to  the  vibration  galvanometer  V. 

With  the  switches  set  so  that  the  dry  cell  and  ordinary  gal- 
vanometer are  connected  to  the  bridge  a  balance  may  be  found 
in  the  usual  way.     Then 
(19)  1*1  =  ?*. 


252      MEASUREMENT  OF  SELF-INDUCTION    [IX,  §  172 

If  both  switches  are  thrown  over,  connecting  the  alternating 
current  and  the  vibration  galvanometer  to  the  circuit,  it  will 
be  found  that  the  bridge  is  no  longer  in  balance,  as  shown  by 
the  broad  band  of  light  on  the  scale.  The  balance  can  be  re- 


]A.C. 


FIG.  114  (repeated). 

stored,  however,  by  rotating  the  -movable  coil  of  the  variable 
inductance  LI  until  the  time  constants  of  the  arms  ac  and  be 
are  equal.  When  this  has  been  done  we  shall  have 


(20) 

or 

(21) 


— 


Since  the  bridge  balance  was  not  disturbed  by  the  rotation  of 
the  movable  coil  of  Ll9  equations  (19)  and  (21)  may  be  com- 
bined, whence 

(22)  ^5=^?, 

and 

(23)  L2  =  L^. 

173.  Laboratory  Exercise  XXXIX.  To  measure  a  self- 
inductance  by  comparison  with  a  variable  standard.  Vibration 
galvanometer  method. 


IX,  §  173J     THE  VIBRATION  GALVANOMETER  253 

APPARATUS.  Variable  inductance  standard,  inductance  to 
be  measured,  vibration  galvanometer,  portable  galvanometer, 
two  resistance  boxes,  two  tap  keys,  two  double-pole  double- 
throw  switches,  and  one  or  two  dry  cells. 

PROCEDURE.  (1)  Arrange  the  circuit  as  in  Fig.  114.  Bal- 
ance the  bridge  in  the  usual  way,  using  a  direct  current. 
Change  to  alternating  current,  and  watch  for  the  broad 
band  of  light  on  the  scale  to  reduce  to  a  sharp  line  as  LL 
is  rotated  from  one  end  of  its  range  to  the  other.  Set 
accurately  on  the  position  for  which  the  vibration  of  the 
galvanometer  coil  disappears  and  read  both  ends  of  the 
pointer. 

Record  this  position  both  in  degrees  and  in  henrys,  and  also 
note  and  record  the  range  through  which  LI  can  be  turned 
without  causing  an  appreciable  broadening  of  the  line  of  light. 
Repeat  several  times,  taking  precaution  to  avoid  being  prej- 
udiced by  previous  settings. 

(2)  Calculate  the  value  of  Z2  from  equation  (23)  and  ex- 
press the  result  in  henrys  and  in  millihenrys.  Indicate  also 
the  probable  precision  of  the  value  found. 


The  vibration  galvanometer  will  be  adjusted  by  an  instructor  to 
resonance  with  the  available  alternating  current  circuit,  and  these  adjust- 
ments should  not  be  changed  by  the  student.  A  fixed  resistance  will  be 
permanently  introduced  between  the  alternating  current  terminals  and 
the  switch  W.  An  adjustable  resistance  which  is  under  the  control  of 
the  student  will  be  introduced  at  r.  This  should  be  set  at  a  large  value 
at  first  and  only  reduced  as  may  be  required. 

In  order  to  extend  the  range  of  the  method,  it  is  sometimes  necessary 
to  include  in  series  with  L\  a  box  of  standard  inductance  coils,  which  can 
be  adjusted  to  several  different  values  by  means  of  plugs  or  switches. 
The  range  of  the  variable  standard  should  be  somewhat  greater  than  the 
smallest  value  of  this  set  of  coils,  thus  providing  a  fine  adjustment  for 
the  fractional  parts  of  the  lowest  box  value.  These  boxes  of  standard 
inductance  coils  are  constructed  so  that  different  values  may  be  obtained, 
and  at  the  same  time  compensating  non-inductive  coils  are  automatically 
introduced  so  that  the  resistance  remains  constant. 


254      MEASUREMENT  OF  SELF-INDUCTION    [IX,  §  174 

174.  Comparison  of  Self-Inductances.  Secohmmeter 
Method.  When  a  source  of  alternating  current  is  not  at 
hand,  a  direct  current  from  any  battery  cell  may  be  made 
available  by  means  of  the  secohmmeter.  This  consists  of  two 
two-piece  commutators  mounted  side  by  side  upon  the  same 
shaft,  each  provided  with  four  brushes  set  90°  apart,  the  shaft 
being  rotated  by  an  electric  motor  or  by  some  other  suitable 
mechanism. 

Figure  115  shows  the  arrangement  of  the  circuit.  As  the 
commutator  B.  C.  to  which  the  battery  is  connected  is  rotated, 
a  pulsating  reversed  current  is  impressed  at  the  bridge  ter- 
minals ab.  Since  an  ordinary  galvanometer  does  not  respond 
to  these  reversed  pulses,  the  commutator  G.  C.  is  connected  to 
the  galvanometer  and  bridge  terminals  as  shown.  This  is 
rotating  with  the  same  speed  as  B.  C.,  the  reversed  current 


FIG.  115. 

pulses  are  rendered  unidirectional  through  the  galvanometer, 
and  any  lack  of  balance  of  the  bridge  is  at  once  indicated  by  a 
steady  deflection.  The  experimental  procedure  is  similar  to 
that  of  Laboratory  Exercise  XXXIX,  §  173,  and  the  value  of 
L  is  calculated  from  equation  (23). 


IX,  §  175] 


GENERAL  INVESTIGATION 


255 


FIG.  116. 


PART  II.     MUTUAL  INDUCTANCE 

175.  Laboratory  Exercise  XL.  A  study  of  mutual  induc- 
tance. 

APPARATUS.  A  rheostat,  resistance  box,  one  or  two  con- 
stant battery  cells,  milliammeter,  two  tap  keys,  ballistic  gal- 
vanometer, and  a  pair  of  coils  for  testing. 

PROCEDURE.  (1)  Arrange  the  circuit  as  in  Fig.  116,  in 
which  E  is  a  control  rheostat,  R1  is  an  adjustable  resistance 
readable  to  tenths  of  an  * 

i      » —  •  *     — 

ohm,  A  is  a  milliammeter,  *j 

and  P  and  S  are  the  two  /T\ 

\.    ) 
coils  of  which  the  mutual     y' 

inductance  is  to  be  studied. 

(2)  Close    K2,    keep    Rl 
constant,  and  vary  E,  ob- 
serving for  several  values  of  the  current  strength  the  corre- 
sponding galvanometer  deflections  when  K^  is  opened  or  closed. 

Tabulate  the  readings  and  plot  a  curve  with  current 
strengths  as  abscissas  and  deflections  as  ordinates.  This  curve 
will  be  a  straight  line  through  the  origin,  showing  that  the 
quantities  induced  in  the  secondary  are  directly  proportional 
to  the  primary  currents. 

(3)  Keep  a  constant  value  of   the  current    through  P  and 
vary  Rf,  observing  several  deflections  as  K±  is  closed. 

It  is  obvious  that  the  damping  in  the  secondary  circuit  will  change  as 
the  resistance  of  the  circuit  changes.  If  a  moving  needle  type  of  galva- 
nometer is  used,  for  which  the  damping  is  small,  the  correction  for  damp- 
ing may  be  found  and  applied  to  the  deflections. 

If  the  moving  coil  galvanometer  is  used,  the  damping,  which  may  be 
large,  is  nearly  all  due  to  induced  currents,  since  the  secondary  circuit  is 
kept  closed.  Changes  in  the  resistance  of  the  secondary  circuit  will  then 
produce  changes  in  the  strength  of  these  induced  currents,  arid  hence,  also 
in  the  damping.  It  follows  that  the  galvanometer  throws  will  be  affected 
by  a  variable  damping,  for  which  corrections  are  not  easily  made.  (See 
§  147.) 


256  MUTUAL  INDUCTANCE  [IX,  §  175 

In  order  to  avoid  this  variable  damping,  it  is  necessary  to  replace  the 
two  tap  keys  by  a  double  key  which  is  so  arranged  that  the  secondary 
circuit  is  broken  a  very  short  time  after  the  primary  is  closed,  thus  allow- 
ing the  galvanometer  to  swing  on  open  circuit  with  a  constant  damping. 
The  time  interval  between  the  make  and  break  should  be  very  short  as 
compared  with  the  time  of  swing  of  the  coil. 

Using  the  double  key  just  described,  take  several  throws  for 
different  values  of  Rf.  Plot  a  curve  with  reciprocals  of  total 
secondary  resistances  as  abscissas,  and  throws  as  ordinates. 
This  should  be  a  straight  line  passing  through  the  origin,  show- 
ing that  the  quantities  induced  are  inversely  proportional  to  the 
secondary  circuit  resistance. 

(4)  Keep  the  primary  current  constant  and  vary  the  distance 
between  the  coils,  reading  the  galvanometer  throws  for  several 
positions.     Turn  S  through  varying  angles  about  a  vertical 
axis,  keeping  the  distances  between  centers  of  P  and  S  con- 
stant, and  for  a  constant  current  in  P  read  throws  on  the  gal- 
vanometer when  KI  is  opened  or  closed. 

(5)  If  possible,  change  the  number  of  turns  on  Pand  on  S 
and  read  the  galvanometer  throws  as  in  (2).     Compare  these 
with  the  throws  as  in  (2). 

(6)  Lay  a  piece  of  iron  lengthwise  through  both  coils,  take 
current  values  and  throws  as  in  (2),  and  compare  the  throws. 

(7)  Set  an  iron  plate  between  the  coils  and  repeat  (6). 

It  will  be  clear  from  the  foregoing  that  the  quantity  induced  in  S  is 
proportional  directly  to  the  primary  current  strength,  inversely  to  the 
secondary  resistance,  and  dependent  also  in  some  way  upon  the  number 
of  primary  and  secondary  turns,  the  size,  shape,  and  relative  positions  of 
the  two  coils  and  the  permeability  of  the  medium.  For  the  particular 
pair  of  coils  given  suppose  that  all  these  factors  except  the  first  two  are 
kept  constant ;  then  we  have 

(24)  Q=^ 

MS 

where  M  is  a  general  constant  which  contains  these  factors,  and  E&  is  the 
total  resistance  of  the  secondary  circuit.     We  may  then  write 

(25)  M 


IX,  §  176]       THE  CAREY   FOSTER  METHOD 


257 


We  have  learned,  however,  that 

(26) 

whence 

(27) 


B. 


which  is  the  definition  for  mutual  inductance  given  in  §  130. 

It  must  be  clearly  understood  that  M  is  independent  of  the  value  of 
the  current  flowing,  unless  iron  is  present  in  the  coils,  and  it  is  dependent 
solely  upon  the  geometry  of  the  circuit,  that  is,  upon  the  dimensions, 
turns,  and  space  relations  of  the  coils.  In  the  report  state  fully  all  the 
inferences  to  be  drawn  from  each  step  of  the  experimental  work. 

176.  Mutual  Inductance  in  Terms  of  a  Capacity.  The 
Carey  Foster  Deflection  Method.  With  a  circuit  arranged 
as  in  Fig.  117,  the  closing  of  the  key  K  establishes  a  current 
of  strength  i  through  the 
primary  coil  P,  and  the 
quantity  Ql  induced  in  the 
secondary  circuit  is  given 
by  the  equation 

/oo\      s~\  JVJ.1  /~i  j 

(AO)     tyi  = 


In  this  equation  c^  is  the  throw  of  the  ballistic  galvanometer, 
G  is  its  constant,  M  is  the  mutual  inductance  of  P  and  S,  and 
s,  r,  and  g  are  the  respective  resistances  of  the  parts  of  the 
secondary  circuit. 

Let  the  circuit  be  changed  to  that  represented  in  Fig.  118, 
where  a  condenser  of  capacity  C  is  placed  in  series  with  the  same 
galvanometer,  and  charged  across  the  resistance  R.  When  K 

is  closed,  the  current  i  flows 
through  E  and  the  charging 
potential  applied  to  the  con- 
denser is  iR.  Then  the  quan- 
K-  3  ^^  £*ven  to  ^e  con(lenser  is 

' —  h|  I "^ ^  given  by 

FIG.  118.  (29)       Qz  =  CRi  =  Gd2, 


258 


MUTUAL  INDUCTANCE 


[IX,  §  177 


where  d2  is  the  galvanometer  throw.     Dividing  (28)  by  (29), 
we  have 

«!  =  _          M        _  =  4 

Q2      CR(s  +  r  +  g)      d2 

whence 


(30) 


(31) 


d/ 


It  is  seen  from  the  figure  that  in  the  first  case  the  galvanom- 
eter swings  on  a  closed  circuit,  of  which  the  resistance  is,  in 
general,  not  high.  In  the  second  case,  however,  the  galvanom- 
eter swings  with  practically  an  infinite  resistance  in  series 
with  it.  For  a  moving  needle  galvanometer  the  deflections  c?x 
and  d2  are  replaced  by  the  sines  of  the  half-angles,  respectively, 
and  the  corrections  for  damping  must  be  applied.  The  value 
of  M  is  then  calculated  from  equation  (31).  For  a  moving 
coil  galvanometer  there  is  so  great  a  variation  in  the  damping 
in  the  two  cases  that  it  is  necessary  to  use  the  zero  method  of 
§  177. 

177.  Mutual  Inductance  in  Terms  of  a  Capacity.  The 
Carey  Foster  Zero  Method.  The  circuits  of  Figs.  117  and 
118  may  be  combined  as  shown  in  Fig.  119.  This  circuit  may 

be  considered  as  made  up 
of  three  parts,  the  primary 
inductive  circuit  a,  the 
capacity  circuit  6,  and 
the  secondary  circuit  c. 
When  K  is  closed,  the 
quantity  given  to  the  con- 


FIG.  119. 


denser  is 

(32)         Q  =  CiR. 


Since  the  galvanometer  may  be  regarded  as  shunted  by  the 
resistance  (s  +  r),  that  part  of  Q  which  goes  through  the  gal- 
vanometer is 


IX,  §  178]      THE  CAREY   FOSTER  METHOD  259 

(33)  Q^CiR     s  +  r    • 

s  +  r  -f  g 

Moreover,  when  K  is   closed,  the   mutual   inductance   sends 
through  the  galvanometer  a  charge  whose  amount  is 


If  adjustments  of  R,  C,  and  r  are  made  such  that  Ql  and  Q2 
are  equal,  it  follows  that  we  may  write 


(35) 

s+r  +g     s+r+g 

which  gives  for  the'Value  of  M, 

(36)  M=Cr(s  +  r). 

By  the  use  of  this  equation  and  the  method  described  above, 
the  galvanometer  deflections  are  always  zero,  the  galvanometer 
resistance  does  not  enter,  and  all  error  due  to  variable  damping 
is  eliminated. 

178.  Laboratory  Exercise  XLI.  To  determine  a  mutual 
inductance  in  terms  of  a  capacity.  Tlie  Carey  Foster  zero 
method. 

APPARATUS.  Standard  adjustable  condenser,  two  resistance 
boxes,  sensitive  moving  coil  ballistic  galvanometer,  one  or  two 
constant  battery  cells,  tap  key  and  mutual  inductance. 

PROCEDURE.  (1)  Set  R  (Fig.  119)  at  some  random  value, 
open  the  secondary  circuit  at  some  point  say  p,  and  tap  /f, 
noting  the  throw  which  occurs.  This  throw  is  due  solely  to 
the  charge  passing  into  the  condenser.  Close  the  circuit  at  p, 
break  the  condenser  circuit  at  some  point  E,  and  note  the 
throw  when  K  is  closed.  This  is  due  solely  to  the  charge  in- 
duced in  S  from  P.  These  two  throws  must  be  in  opposite 
directions.  If  they  are  not,  interchange  the  terminals  of  P. 

(2)  With  all  connections  as  shown  in  Fig.  119,  make  r  equal 
to  zero,  tap  K,  and  note  the  throw.  Make  r  infinite,  that  is, 


260  MUTUAL  INDUCTANCE  [IX,  §  178 

open  the  circuit  at  p  ;  again  tap  K  and  note  the  throw.  If  the 
two  throws  are  in  opposite  directions,  a  value  can  be  found  for 
r  for  which  no  deflection  will  occur. 

If  the  two  deflections  are  not  in  opposite  directions,  then 
the  condenser  is  sending  more  quantity  through  the  circuit 
than  is  induced  in  S,  and  no  value  of  r  whatever  can  be  found 
for  which  there  will  be  no  deflection.  The  quantity  from  the 
condenser  must  then  be  reduced,  either  by  making  C  smaller 
or  by  diminishing  the  charging  potential  across  its  terminals. 
It  is  well  not  to  change  the  value  of  R  because  that  would 
alter  the  current  through  P.  The  charging  potential  is  most 
conveniently  varied  by  means  of  a  traveling  contact  at  E, 
which  enables  C  to  be  charged  across  any  desired  fraction  of  R. 

(3)  Having  found  a  combination  of  C,  R  and  r  for  which  the 
deflection  is  zero,  record  these  values  together  with  the  least 
amount  by  which  r  can  be  increased  or  diminished  before  the 
smallest  perceptible  deflection  occurs.     Repeat  the  adjustment 
of  r  several  times,  with  the  attention  fixed  on  the  galvanometer 
and  with  no  prejudice  from  previous  settings.     Approach  the 
final  value  of  r  both  from  values  that  are  too  high  and  those 
that  are  too  low.     Find  one  or  two  other  combinations  of  R, 
C,  and  r,  and  repeat  the  readings  as  outlined  above. 

(4)  Calculate  the  value  of  M  from  equation  (36),  and  state 
the  probable  precision  of  the  result. 

(5)  State  in  the  report  the  reasons  for  all  the  steps. 

If  the  values  of  (7,  .Z?,  and  r  are  given  in  absolute  units,  M  will  be  in 
centimeters.  If  the  capacity  is  in  farads,  and  the  resistances  are  in  ohms, 
then  M  will  be  in  henrys.  If  C  is  in  microfarads,  then 

(37)  M  =  CR(s  +  r)  10~6  henrys. 

179.  Comparison  of  Two  Mutual  Inductances.  Maxwell's 
Method.  Two  mutual  inductances  to  be  compared  are 
arranged  in  a  circuit  as  shown  in  Fig.  120.  A  constant  current 
flows  through  P  and  P  when  K  is  closed.  In  series  with  the 


IX,  §  179] 


MAXWELL'S  METHOD 


261 


secondary  coils  are  resistance  boxes  r  and  r'.  When  K  is 
tapped,  considering  each  side  of  the  circuit*  separately,  the 
charges  induced  are 

(38)  Q  =  - 


(39) 


If  we  consider  charges  induced  in  both  sides  of  the  circuit 
at  the  same  time,  however,  and  if  values  of  r  and  r1  are  so 
adjusted  that  Q  =  Q',  there  will  be  no  resultant  charge  through 
the  galvanometer 
and  hence  no  de- 
flection. 

In  general, 
there  are  two 
charges  passing 
through  the  gal- 
vanometer, in  op- 
posite directions, 
and  of  different 
values.  If  the 
above  condition  is  realized,  so  that  Q  =  Qf,  these  charges 
exactly  annul  each  other  between  c  and  d.  This  means  that 
charge  flows  away  from  c  on  one  side  as  fast  as  it  is  supplied 
on  the  other  side,  and  the  potentials  at  c  and  d  remain  con- 
stant. The  galvanometer  resistance  then  has  no  effect  on  the 
induced  charges  and  g  may  be  dropped  from  (38)  and  (39). 
Solving  these  equations  for  M2,  which  may  be  taken  as  the 
unknown,  we  have 

(40) 


a 
""^^^n 

c 

p 

\          r 

A    , 

0 

8'                  1 

1  —  iRfiOTRfiN 

d 

U*   '  — 

LMRRlOOftOfP  —  ! 

P' 

ill 

K 

P 
^           , 

FIG.  120. 


. 

The  time  constants  of  the  two  parts  of  the  secondary  circuit 
should  be  nearly  or  quite  alike,  and   a  long-period   ballistic 


262  MUTUAL  INDUCTANCE  [IX,  §  180 

galvanometer  should  be  used,  so  that  the  time  of  throw  is 
large  as  compared  with  the  time  constant  of  either  side. 
Increased  sensibility  may  be  had  by  using  an  alternating  cur- 
rent and  a  vibration  galvanometer. 

180.  Laboratory  Exercise  XLII.  To  compare  two  mutual 
inductances.  Maxwell's  method. 

APPARATUS.  A  standard  mutual  inductance  and  one  to  be 
measured,  sensitive  ballistic  galvanometer,  two  resistance 
boxes,  one  or  two  constant  battery  cells,  tap  key,  and  box 
bridge. 

PROCEDURE.  (1)  Arrange  the  circuit  as  in  Fig.  120.  Open 
the  circuit  at  p  and  tap  K,  noting  the  throw,  which  is  due 

p  to  the  charge  in- 


vQQQQQQJM). 
P 


H4> 


K  R 
V*^/X/V 


duced  in  s'.  Close 
p,  break  the  cir- 
cuit at  a,  tap  K, 
and  note  the 
throw  which  is 
due  to  the  charge 
induced  in  s  only. 
These  throws 

must    be    in    op- 
Fia.  120  (repeated) . 

posite  directions. 

(2)  With  connections  as  shown,  adjust  r  and  r'  until  no 
deflection  occurs.     Find  the  resistances  of  the  coils  s  and  s' 
with  the  box  bridge. 

(3)  Take  different  values  of  the  current  through  R  and  dif- 
ferent settings  of  r  and  r'.     Record  in  the  data  how  much  r  or 
r'  must  be  changed  in  order  to  produce  the  least  perceptible 
deflection. 

(4)  Calculate  the  value  of  M  from  equation  (40)  and  indi- 
cate the  probable  precision  of  the  value  found. 


IX,  §  182]  MAXWELL'S  METHOD  263 

181.  Laboratory  Exercise  XLIII.     Comparison  of  the  values 
of  M  for  a  current  inductor  as  determined  by  the  Carey  Foster 
method  and  by  direct  measurement. 

APPARATUS.     As  in  §  178,  together  with  a  current  inductor. 

PROCEDURE.  (1)  Determine  the  value  of  M  by  the  method 
of  §  178.  Take  a  large  number  of  readings  and  use  great 
care  throughout. 

(2)  From  the  data  furnished,  calculate  the  value  of  M}  using 
equation  (17),  §  131. 

(3)  Compare  the  results  obtained  in  (1)  and  (2)  and  account 
for  their  variation. 

182.  Laboratory  Exercise  XLIV.     To  determine  a  mutual  in- 
ductance with  the  vibration  galvanometer  and  a  variable  standard 
of  self-inductance. 

APPARATUS.     As  in  §  173. 

PROCEDURE.  (1)  Arrange  the  circuit  as  in  Fig.  114.  Con- 
nect the  two  coils  whose  mutual  inductance  is  to  be  found  in 
helping  series  and  call  the  self-inductance  of  this  arrangement 
LI.  Determine  its  value  by  the  method  of  §  173. 

(2)  Eeverse  the  terminals  of  one  coil,  thus  connecting  them 
in  opposing  series,  and  call  the  self-inductance  of  this  arrange- 
ment LI'.     Determine  its  value  as  before. 

(3)  Calculate  the  value  of  M  from  the  relation 


as  given  in  equation  (71),  §  141. 


CHAPTER   X 


-       .      >.<•         .  •« 

'  •./,.."•«/•:  ':•£ 

\^l\Uf/l£f.  '> 

(        9    *    .   •  • '.'»:  '/ 

-.':tf 'j/f 


MAGNETISM   AND   THE   MAGNETIC   CIRCUIT 

183.  Magnetism  Produced  by  Electric  Currents.  That 
condition  of  matter  called  magnetic  can  be  produced  in  certain 
subtances  by  the  influence  of  a  natural  magnetic  iron  ore  Fe304> 

or  by  the  action  of  electric 
currents.  It  is  highly  prob- 
able that  the  magnetic  bodies 
found  in  nature  owe  their 
peculiar  and  characteristic 
properties  in  some  way  to 
electric  discharges.  Hence, 
the  conclusion  may  be  drawn 
that  the  electric  current 
is  the  primary  source  of 
magnetism. 

It  was  stated  in  §128 
that  lines  of  force  are  to 
be  thought  of  as  closed  curves,  which  may  be  established  by 
the  passage  of  a  current  through  a  circuit.  The  lines  of  force 
are  interlinked  with  the  circuit.  Both  the  lines  of  force  and 
the  electric  circuit  are  closed  curves. 

Figure  121  shows  the  appearance  of  the  magnetic  field  about 
a  straight  wire  carrying  a  current,  the  lines  of  force  forming 
concentric  circles  about  the  wire.  The  field  strength  at  any 
distance  r  from  the  wire  is  given  by  equation  (8),  §  105,  and  is 


FIG.  121. 


264 


X,  §  183] 


THE   MAGNETIC   CIRCUIT 


265 


Figure  122  shows  the  field  in  a  plane  at  right  angles  to  the 
plane  of  a  single  circular  coil,  through  the  center  of  the  coil. 

The  strength  of  the   field   at   the         ,„,,.     .'„ ,. 

center  is  given  by  equation   (14), 
§106, 


Figure  123  shows  the  mag- 
netic  field  about  a  solenoid.  The 
strength  of  field  at  the  center  of  a 
long  solenoid  is  given  by  equation 


Not  all  substances  are  suscepti- 
ble  in   the   same    degree    to    this 
magnetic  influence.     Indeed,  most 
substances  are  less  susceptible  than  dry  air  or  a  vacuum ;   to 
this  class  of  substances  has  been  given  the  name  diamagnetic. 


^-Qi^A-  .v\vi 

g&^i 

^<k^\ 
5t^5**> 


Wi%&& 

*.' ,'s  <?'.'. ^-'.1 

7%%M 

'&ZJjMk 

z?^m 


FlQ.  123. 


On  the  other  hand,  those  substances  which  are  more  suscep- 
tible to  magnetism  than  air  are  called  paramagnetic.     Three 


266  MAGNETISM  [X,  §  183 

substances,  iron,  nickel,  and  cobalt,  which  are  chemically  re- 
lated, show  this  phenomenon  in  a  degree  much  greater  than 
any  other.  Since  iron  greatly  exceeds  the  other  elements  in 
this  respect,  the  name  ferromagnetic  is  applied  to  the  group, 
and  this  descriptive  word  is  commonly  understood  even  though 
shortened  to  magnetic.  With  the  exception  of  certain  alloys, 
briefly  discussed  later,  the  various  forms  of  iron  and  steel  are 
the  only  magnetic  materials  considered  in  the  following  pages. 

184.  The  Magnetic  Circuit.  It  has  been  shown  in  §  109 
that  for  a  short  distance  at  the  center  of  a  long  solenoid  the 
magnetic  field  is  very  nearly  uniform.  If  the 
solenoid  is  bent  into  the  form  of  a  circle,  with 
the  ends  joined,  the  effect  of  the  ends  will 
vanish  and  a  nearly  uniform  field  will  exist 
everywhere  within  the  windings. 

Consider  a  ring   of  iron  of  circular   cross- 
section,  in  the  form  of  the  tore  or  anchor  ring, 
uniformly  overwound  with  wire  turns  through  which  a  current 
flows.     This  constitutes  an    v\ <  ••-.- •'••VT-^-     '-.-:   *•*•**  -"-..•-' 

*.»•*.  •",?•'  *•'"•';: •/•«. .'*;. v,  **—•-.  .-"  ••»'*•  .7     *"•"-••*'••• 

endless  solenoid,  containing  *•:•/•;?  '*m%\e^Gf.}&f^\^^>.f'*?i  ••'/; 
an  iron  instead  of  an  air  S^l-A^"  £v'r^r^£^\j.}  -3?^%  i-i/'\: 

*•»'!-*      •    . .  ^>  j «     •     ..."_'"'    V*       .'   *.     '         ^"^'  -  *  'X-*  **  **   '    . 

core.  Current  through  this 
toroidal  winding  will  pro- 
duce the  uniform  magnetic 
field  described  above,  and 
will  establish  a  magnetic 
state  within  the  iron  which 
is  Avholly  confined  to  the 
iron,  there  being  no  exter- 
nal field. 

Figure     124     represents 
such  a  ring  with  the  mag-  FIG.  125. 

netic  lines  all   within   the   iron.     If  this  ring,   with  such  a 


X,  §  184] 


THE   MAGNETIC   CIRCUIT 


267 


FIG.  126. 


magnetic  state  existing  within  it,  is  placed 

on  a  horizontal  surface  and  covered  with  a 

sheet  of  paper  over  which  iron  filings  are 

sprinkled,  there  will  appear  no  regularity 

in  the  arrangement  of  the  particles,  which 

shows  that  there  is  no  external  effect. 

Figure  125  is  a  picture  of  the  iron  fil- 
ings in  such  a  case.      Although  there   was   within  the   ring 

a  strong  magnetic  flux, 
no  definite  arrangement 
of  the  particles  can  be 
seen. 

A  marked  change 
takes  place,  however,  as 
soon  as  the  magnetic 
circuit  is  interrupted 
by  some  non-magnetic 
substance,  such  as  air. 
The  same  ring  used 
for  Fig.  125  was  cut 
through  on  one  side 

with  a  metal  saw,  making  a  gap  one  tenth  of  a  millimeter  wide, 

as  shown  .at  A,  Fig.  126. 

A  distinct  regularity  of 

arrangement  of  the  par- 
ticles is  now  evident,  as 

shown  in  Fig.  127.     If 

the  air  gap  is  increased 

to    one    centimeter,   as 

shown  in  Fig.  126,  B,  a 

still  greater  effect  is  ob- 

i  •     ,-,  •--  -  -."•&••'*• 

served  in  the  space  sur-    :-  ':•  ^  fss's  •' 

rounding    the    ring,   as 

shown  in  Fig.  128.  FIG.  128. 


FIG.  127. 


268 


MAGNETISM 


[X,  §  184 


If  the  air  gap  is  still  further  enlarged  the  piece  takes  the 
form  of  the  ordinary  horse-shoe  magnet,  and  in  this  case  the 


FIG.  129. 


arrangement  of  the  iron  filings  is  shown  in  Fig.  129.     This 
magnet  is  one  taken  from  an  ordinary  telephone  magneto.     If 


FIG.  130. 

the  ends  are  bent  still  further  back,  the  iron  particles  group 
themselves  as  in  Fig.  130. 

Two  conclusions  may  be  drawn  from  a  careful  study  of  the 
preceding  cases  :  (1)  lines  of  force  appear  to  be  closed  curves  ; 
(2)  magnetic   circuits  may  be  divided   into   two  classes,  per-f 
feet  and  imperfect. 


X,  §  185] 


THE  MAGNETIC  CIRCUIT 


269 


V 


FIG.  131. 


185.  Perfect  and  Imperfect  Magnetic  Circuits.  The  per- 
fect magnetic  circuit  may  be  defined  as  one  in  which  the  mag- 
netic material  is  continuous  and  homogeneous,  and  about 
which  no  external  magnetic  field  exists,  the  magnetic  flux 
being  wholly  confined  to  the  material  of  which  the  circuit  is 
composed.  Such  perfect  magnetic  circuits  are  rare.  It  is 
only  approximately  realized  in  dynamo 
and  motor  frames,  Figs.  131  and  132. 

The  imperfect  magnetic  circuit  is  one 
in  which  the  body  of  the  magnetic  ma- 
terial is  interrupted  for  a  greater  or  less 
portion  of  its  length  by  some  non-mag- 
netic material.  It  is  characterized  by 
the  external  field  about  it,  in  which  a 
force  is  observed  to  act  upon  a  magnetic 
pole  placed  in  it.  Most  magnetic  circuits  are  of  this  type. 
This  straying  of  the  magnetic  field  outside  of  the  material 
of  the  circuit  is  called  magnetic  leakage.  This  leakage  varies 
from  zero  in  the  case  of  the  ring  to  a  maximum  in  the  case 
of  the  short  bar  magnet.  Leakage  varies  over  the  surface  of 

a  magnet,  being  greatest  opposite  two  points 

near  the  ends  of  the  bar,  and  least  at  the 
middle  of  the  bar,  as  shown  in  Fig.  130. 

When  a  straight  bar  magnet  is  pivoted  at 
its  center,  so  that  it  is  free  to  swing  in  a 
horizontal  plane,  it  always  sets  its  magnetic 
axis  parallel  to  the  direction  of  the  field  in 
which  it  is  placed.  This  magnetic  axis  is 
usually  its  longest  axis. 

The  behavior  of  the  bar  is  now  precisely  that  which  it 
would  be  if  its  magnetism  were  concentrated  in  two  points 
very  near  the  ends  of  the  bar.  These  two  hypothetical  points 
are  called  the  poles  of  the  magnet.  However,  it  can  be 
shown  experimentally  that  this  magnetism  is  not  concen- 


FTG.  132. 


270  MAGNETISM  [X,  §  185 

trated  at  certain  points,  but  is  distributed  along  the  surface 
of  the  bar. 

Magnetic  polarity  is  developed  wherever  lines  of  force  enter 
or  leave  the  surface.  The  number  of  lines  of  force  per  square 
centimeter,  that  is,  the  surface  density,  is  a  measure  of  the 
distributed  pole  strength. 

For  a  magnet  floated  on  water  or  otherwise  freely  sus- 
pended, there  is  no  motion  of  translation,  simply  rotation. 
Since  the  earth's  field  is  uniform  for  the  limited  region  about 
the  magnet,  this  proves  that  the  resultant  turning  moments 
acting  on  the  two  ends  of  the  bar  are  equal  and  opposite. 

A  survey  of  the  foregoing  facts  and  phenomena  leads  to  the 
fundamental  principle  that  magnetism  is  a  circuital  phenom- 
enon. Like  electricity  in  motion,  or  an  incompressible  fluid 
flowing  through  a  pipe,  it  cannot  flow  in  one  portion  of  the 
circuit  only.  Magnetic  polarity  is  developed  only  when  the 
material  of  the  magnetic  circuit  abruptly  changes  its  per- 
meability. 

186.  Specification  of    Magnetic   Quantities.      There    are 
two  methods  of  expressing  the  values  of  magnetic  quantities. 

In  the  first  method  an  imperfect  magnetic  circuit  is  assumed 
which  has  free  poles  and  an  external  field,  and  the  magnetic 
condition  is  specified  in  terms  of  the  force  action  on  a  mag- 
netic pole  when  placed  in  this  field. 

In  the  second  method  the  magnetic  state  is  specified  in 
terms  of  analogies  with  the  flow  of  fluids,  or  of  electric 
currents. 

Definitions  of  certain  magnetic  units  by  the  first  method  are 
given  in  §§  187-189.  Definitions  of  other  units  by  the  second 
method  are  given  in  §§  190-191.  These  units  are  compared  in 
§  192. 

187.  Magnetic  Field  Strength.     A  magnetic  field  is  com- 
pletely specified  by  its  action  on  the  unit  pole  placed  in  it. 


X,  §  189]  THE   MAGNETIC   CIRCUIT  271 

The  direction  of  the  field  is  given  by  the  direction  in  which  a 
free  north-seeking  pole  will  move.  The  intensity  or  strength 
of  the  field  is  given  by  the  force  in  dynes  which  acts  on  a  unit 
pole.  Magnetic  field  strength  is  therefore  expressed  in  dynes 
per  unit  pole.  It  is  usually  represented  by  the  symbol  H. 

188.  Magnetic  Moment.     It  is  convenient  to  think  of  the 
magnetic  poles  as  being  located  at  definite  points  within  the 
magnetic  substance.     In  reality,  however,  the  entire  body  of 
the    magnet    possesses    this    polarity    in    a    greater    or    less 
degree.     If  a  bar  magnet  is  hung  in  a  horizontal  position  at 
right  angles  to  a  horizontal  magnetic  field  of  unit  strength,  it 
will  tend  to  set  itself  parallel  to  the  field,  and  the  moment  of 
the  force  couple  which  tends  to  rotate  it  is 

(1)  M=ml, 

where  m  is  the  pole  strength  and  /  is  the  distance  between  the 
poles.  The  quantity  M,  which  is  called  the  magnetic  moment 
of  the  bar,  is  really  the  sum  of  all  the  moments  acting,  con- 
sidering that  the  distributed  polarity  has  a  different  lever  arm 
for  every  point  along  the  bar. 

189.  Intensity  of  Magnetization  and  Susceptibility.    If 

the  pole  strength  of  the  magnet  is  divided  by  the  area  of  the 
cross-section,  the  quotient  gives  the  intensity  of  magnetization. 
It  is  represented  by  the  symbol  I,  and  its  value  is  given  by  the 
equation 

(2)  |  =  * 

where  m  denotes  the  pole  strength  and  a  denotes  the  area  of 
the  cross-section. 

If  both  numerator  and  denominator  are  multiplied  by  the 
length  of  the  specimen,  we  have 

/o\  i      ml     M 

3  I  =      = 


272  MAGNETISM  [X,  §  190 

hence  I  may  be  defined  also  as  the  magnetic  moment  per  unit 
of  volume. 

When  a  piece  of  iron  is  placed  in  a  magnetic  field  of  strength 
H,  the  inductive  action  of  the  field  develops  poles  in  the  bar, 
and  gives  it  an  intensity  of  magnetization  I. 

The  ratio  of  I  to  H  is  called  the  susceptibility,  and  it  is 
represented  by  the  symbol  k.  We  may  then  write 

(4)  *-i. 

190.  Magnetic  Flux.  The  arrangement  of  iron  filings  in  a 
magnetic  field  suggests  that  the  lines  of  force  are  continuous 
closed  curves.  It  is  often  convenient  to  regard  them  as  analo- 
gous to  stream  lines  in  a  moving  fluid.  From  this  view  point 
the  expression  magnetic  flux  has  a  definite  meaning,  and  mag- 
netic phenomena  may  be  treated  as  circuital  in  type,  the 
magnetic  circuit  having  laws  and  properties  analogous  to  those 
of  the  electric  circuit.  (See  §  193.) 

The  magnetic  flux  is  to  be  considered  as  distributed  more  or 
less  uniformly  throughout  the  entire  magnetic  circuit.  This 
distribution  is  represented  by  lines  drawn  closer  together 
where  the  field  is  strong,  and  farther  apart  where  it  is  weak. 
Any  definite  portion  of  this  field  may  be  thought  of  as  marked 
off  from  neighboring  portions  by  the  walls  of  an  imaginary 
tube.  By  properly  selecting  the  diameter  of  this  tube  we  may 
arrive  at  a  definition  of  the  unit  flux. 

We  will  select  a  cross-section  for  this  imaginary  bundle  of 
stream  lines  such  that  a  conductor  moving  across  it  in  one 
second  will  develop -a  potential  difference  of  one  absolute  unit. 
This  definite  amount  of  magnetic  flux  is  called  the  line  of 
force,1  or  the  maxwell.  The  maxwell  is  then  defined  as  the 

1  The  expression  line  of  force  has  two  quite  distinct  meanings.  It  is  some- 
times used  to  signify  the  direction  of  the  field  at  a  given  point,  and  sometimes 
it  is  used,  as  in  the  case  here  cited,  to  mean  the  unit  amount  of  magnetic  flux. 


X,  §  191]  THE   MAGNETIC   CIRCUIT  273 

magnetic  flux  which  a  single  conductor  must  cut  in  one  second 
in  order  to  develop  one  absolute  unit  of  potential  difference.1 

If  the  area  over  which  this  magnetic  flux  is  distributed  is 
enlarged,  the  conductor  must  move  faster  in  order  to  cut  the 
required  amount  in  one  second.  This  shows  the  necessity  for 
another  quantity,  the  flux  density.  The  unit  of  flux  density 
is  called  the  gauss:  a  magnetic  field  is  said  to  have  a  strength 
of  one  gauss  when  there  is  a  uniform  distribution  of  one 
maxwell  of  flux  over  an  area  of  one  square  centimeter  taken  at 
right  angles  to  the  direction  of  the  flux. 

The  total  flux  is  usually  represented  by  the  symbol  <£,  and 
the  flux  density  in  air  by  the  symbol  H.  For  a  uniform  dis- 
tribution over  an  area  A  we  may  then  write 

(5)  <£  =  HA 

191.  Induction  Density  and  Permeability.  When  a  bar  of 
iron  is  placed  in  a  magnetic  field  the  magnetic  flux  through  it 
is  very  greatly  increased,  and  is  then  expressed  in  terms  of 
lines  of  induction,  the  expression  lines  of  force  being  restricted 
solely  to  the  inducing  field.  The  total  flux  of  lines  of  induc- 
tion in  the  bar  is  represented  by  <£,  and  the  flux  density  of  the 
induction  by  B,  then 

(6)  4>  =  BA, 

where  A  is  the  cross-section  of  the  iron  bar.  As  in  the  case  of 
air,  the  unit  of  induced  magnetic  flux  is  called  the  maxwell; 
and  the  unit  of  induction  flux  density,  or  one  line  of  induction 
per  square  centimeter,  is  for  practical  purposes  called  the 
gauss. 

If  B  represents  the  induction  density  in  the  iron  after  being 
placed  in  a  magnetizing  field  of  strength  H,  the  ratio  of  B  to 

1  If  108  maxwells  are  cut  by  a  single  conductor  in  one  second,  the  induced 
potential  difference  is  one  volt.  There  is  no  generally  accepted  name  for  this 
larger  flux  unit,  although  it  is  sometimes  called  the  practical  line,  in  contra- 
distinction to  the  C.  G.  S.  line  or  maxwell. 


274  MAGNETISM  [X,  §  192 

H  is  called  the  permeability,  and  it  is  represented  by  the 
symbol  /x,.  The  permeability  is  given  by  the  equation 

(7)  ,  =  £• 

The  general  equation  for  magnetic  flux  may  be  written 

(8)  *  =  pUA. 

This  reduces  to  equation  (5)  when  the  flux  is  through  air,  for 
which  ^  =  1.  When  the  flux  is  through  iron  it  is  identical 
with  equation  (6). 

192.  Comparison  of  Magnetic  Quantities.    When  a  sphere 
of  unit  radius  is  drawn  about  the  unit  pole  as  a  center,  there 
is  at  every  point  of  the  surface  a  unit  field  strength  of  one 
dyne  per  unit  pole.     The  flux  which  exists  through  each  square 
centimeter  of  surface  is  one  maxwell.     Since  the  area  of  the 
unit  sphere  is  4  ?r  square  centimeters,  4  TT  maxwells  is  the  total 
flux  from  the  unit  pole. 

If  the  unit  pole  is  replaced  by  one  of  strength  m  units,  the 
total  flux  is  given  by 

(9)  <£  =  47rm. 

In  comparing  the  two  methods  of  specifying  magnetic  quan- 
tities just  given,  it  will  be  seen  that  the  intensity  of  magneti- 
zation I  is  a  measure  of  a  condition  which  can  be  produced  in 
magnetic  materials  alone.  Magnetic  flux  <f>  is  more  general  and 
is  descriptive  of  a  condition  which  can  exist  in  any  substance, 
whether  magnetic  or  not.  Every  known  substance  can  have 
magnetic  flux  established  in  it  by  a  magnetizing  field ;  hence 
an  insulator  for  magnetism  is  unknown.  The  field  strength  H 
may  be  regarded  as  the  cause  of  both  I  and  <j>. 

193.  The  Law  of  the  Magnetic  Circuit.    Magnetomotive 

Force.  The  analogy  between  the  flow  of  a  fluid  through  a 
pipe  and  electric  current  through  a  circuit  was  extended  to 


X,  §  193]  THE   MAGNETIC   CIRCUIT  275 

the  magnetic  circuit  as  early  as  1871.  The  electromotive  force 
has  been  defined  in  terms  of  the  work  done  in.  moving  a  unit 
charge  once  around  a  complete  circuit,  and  a  similar  expression 
for  the  work  done  in  moving  a  unit  pole  once  around  a  mag- 
netic circuit  along  a  line  of  force  was  called  by  Maxwell  the 
line  integral  of  the  magnetic  force.  Bosanquet  later  called  this 
the  magnetomotive  force,  and  expressed  the  relations  of  the 
magnetic  circuit  in  a  formula  similar  to  Ohm's  law. 

(10)  magnetic  flu*  =  magnetomotive  force 

reluctance 

The  work  done  dw  in  moving  a  unit  pole  along  any  line  of 
force  is  measured  by  the  product  of  the  length  dL  of  the  path 
by  the  component  of  the  magnetic  force  along  this  path,  or 

(11)  dw=UcosOdL 

and  the  total  work  is  found  by  integrating  this  expression 
around  this  entire  path,  whatever  its  form  may  be.  This 
principle  is  applicable  to  any  path  whatever  in  the  magnetic 
field.  If  the  path  coincides  at  every  point  with  a  line  of  force 
or  a  line  of  induction,  however,  0  is  everywhere  zero,  and  the 
line  integral  becomes  simply 

(12)  W 

We  have  already  seen  (§  105)  that  the  work  done  in  moving 
the  unit  pole  once  around  a  single  turn  of  wire  carrying  a 
current  of  strength  i  is  W  =  4  TTI.  If  there  are  N  effective 
turns,  the  line  integral  or  magnetomotive  force  M.  M.  F., 
becomes 

(13)  M.M.F.  =  4irJV»V 
Moreover,  by  equation  (34),  §  109, 


276  MAGNETISM  [X,  §  193' 

whence 

(14)  M.  M.  F.  =  H£  =  4  irNi. 
If  i  is  measured  in  amperes,  we  have 

(15)  M.  M.  F.  =  T%  irNL 

The  absolute  unit  of  M.  M.  F.  is  called  the  gilbert,  and  the 
practical  unit  is  the  ampere-  turn.  A  magnetomotive  force 
expressed  in  gilberts  is  reduced  to  the  equivalent  number  of 
ampere-turns  by  dividing  by  4  Tr/10.  Magnetizing  held  strength 
may  be  expressed,  in  terms  of  gilberts  per  centimeter  or 
per  inch,  or  in  terms  of  ampere-turns  per  centimeter  or  per 
inch. 

Suppose  that  a  bar  of  iron  of  length  L,  placed  within  a 
solenoid,  has  an  induction  density  of  B  established  in  it  by  the 
field  of  strength  H.  Neglecting  the  effect  of  the  field  outside 
the  solenoid,  we  may  write 


or 

M.M.F. 


,1A, 


The  quantity  R  is  called  the  reluctance  of  the  circuit.     (See 
§  194.) 

The  established  flux  is  seen  to  be  directly  proportional  to 
the  M.  M.  F.  ;  hence  the  magnetomotive  force  may  be  regarded 
as  a  measure  of  the  effectiveness  of  the  field  in  magnetizing  the 
piece  of  iron.  The  general  equation  (16)  presumes  an  initial 
state  devoid  of  magnetism,  but  the  flux  established  by  any 
magnetomotive  force  is  somewhat  dependent  upon  any  pre- 
vious magnetic  history  of  the  circuit.. 

194.  Reluctance.  Reluctance  is  a  property  of  the  mag- 
netic circuit  which  resists  magnetization.  It  is  inversely  pro- 


X,  §  194]  THE  MAGNETIC  CIRCUIT  277 

portional  to  the  cross-section  A,  and  directly  proportional  to 
the  length  L  of  the  circuit.  The  factor  p  by  which  L/A  must 
be  multiplied  to  obtain  the  reluctance  is  called  the  specific  re- 
luctance or  reluctivity  of  the  substance.  We  may  then  write 

(17)  R  =  ^ 

The  reciprocal  of  the  reluctivity  is  called  the  permeability,  and 
is  denoted  by  /x  ;  hence  we  have 

SSplpt  '-?  ••"•"   .    ^;*v 

whence 


This  expression  is  seen  to  be  the  denominator  of  the  middle 
term  in  equation  (16). 

The  unit  of  reluctance  is  the  oersted]  which  is  the  reluc- 
tance of  a  circuit  in  which  one  gilbert  establishes  a  flux  of  one 
maxwell.  The  oersted  may  also  be  defined  as  the  reluctance 
of  an  air  gap  one  centimeter  long  and  one  square  centimeter 
in  cross-section. 

Reluctance  and  its  reciprocal  permeance  are  characteristics 
of  the  circuit.  Reluctivity  and  permeability  are  characteristic 
of  the  given  material. 

Ohm's  law  shows  that  resistance  is  independent  of  current 
strength.  The  reluctance,  however,  varies  with  the  magnetic 
flux,  which  in  turn  varies  with  the  permeability,  as  shown  in 
Fig.  147,  so  that  the  analogy  between  Ohm's  law  and  the  law 
for  the  magnetic  circuit  in  equation  (10)  is  not  complete. 
Moreover,  although  energy  is  required  to  establish  or  reduce 
the  magnetic  flux  through  iron,  no  energy  is  required  to  main- 
tain a  continuous  flux.  There  is,  then,  no  analogy  to  Joule's 
law  in  the  magnetic  circuit. 

The  magnetic  circuit  may  not  be  homogeneous,  but  may 
comprise  various  portions  of  different  lengths,  cross-sections, 


278 


MAGNETISM 


[X,  §  194 


and  permeabilities,  including  one  or  more  air  gaps.  Practical 
problems  dealing  with  such  circuits  are  solved  by  rinding  the 
magnetomotive  force  necessary  to  establish  the  desired  flux  in 
each  part  of  the  circuit.  The  total  flux  is  then  given  by  the 
equation 


(19) 


Except  in  special  apparatus  for  testing,  it  is  seldom  that  a 
uniform  winding  can  be  placed  over  the  entire  circuit.  More 

often  the  wire  turns 
constitute  what  is 
called  a  local  or 
bunched  winding. 
The  external  field  due 
to  a  bunched  winding 
on  an  anchor  ring  is 
shown  in  Fig.  133. 

For  a  circuit  such 
as  that  shown  in  Fig. 
131,  the  M.M.F.  of 
the  windings  is  calcu- 
lated for  the  required 
flux  by  equation  (19),  and  corrections  are  applied  for  the 
leakage  at  the  air  gap.  (See  §  198.) 

195.  The  Relation  Between  B  and  I.  When  a  long,  un- 
magnetized  bar  of  iron  of  cross-section  A  is  placed  in  a  uni- 
form field  of  strength  H,  for  example  at  the  center  of  a 
long  solenoid  carrying  current,  the  total  magnetic  flux  in  the 
bar  is  made  up  of  the  sum  of  two  distinct  components  (a)  and 
(6) :  (a)  through  the  space  occupied  by  the  bar  there  will  be 
a  magnetic  flux  due  to  the  original  magnetizing  field  of  value 
\\A ;  (6)  when  the  bar  of  iron  is  placed  in  the  field,  magnetic 


FIG.  133. 


X,  §  197]  THE  MAGNETIC  CIRCUIT  279 

poles  of  strength  m  are  induced,  and  by  equation  (9)  the  flux 
due  to  its  own  poles  is  4  Trra. 

The  total  flux  through  the  iron,  or  the  induction,  is  therefore 

<£  =  HA  +  4  TTW, 
and  the  induction  density  is 

(20)  B  =  ^= 


The  quantity  B  includes  the  effect  of  both  I  and  H,  while  I 
includes  only  the  effect  of  H,  and  not  H  itself. 

196.  The    Relation    Between    k   and  \L.     Substituting  in 
equation  (20)  the  values  of  B  and  I  from  equations  (4)  and  (7), 
we  may  write 

p.H=U+  4,rA:H, 
whence 
(21)  ,1=1  +  4**:. 

For  vacuum,  dry  air,  and  non-magnetic  substances  generally, 
k  =  0  and  ^  =  1.  For  paramagnetic  substances  k  is  positive  and 
/w,  is  greater  than  one.  For  diamagnetic  substances  k  is  nega- 
tive and  /A  is  less  than  one.  However,  there  is  no  known  sub- 
stance for  which  /u,  is  as  much  less  than  one  as  it  is  greater 
than  one  for  ferromagnetic  materials.  For  bismuth,  which 
shows  the  largest  susceptibility  of  any  diamagnetic  substance, 
k  —  —  •  14  x  10~6,  which  corresponds  to  a  permeability  of  0.9998. 

197.  The  Demagnetizing  Effect  of  Poles.    Imperfect  mag- 
netic  circuits   have   a   tendency  to   demagnetize   themselves. 
The  flux   density  within  the  bar  is  given  by  equation   (20). 
The  factor  H  is  not  constant,  however,  since  the  bar  develops 
poles  as  soon  as  the  magnetic  state  is  induced  in  it.     These 
poles  act  throughout  the  space  occupied  by  the  bar,  and,  hence, 
on  the  bar  itself,  and  this  polar  field  will  have  a  direction  oppo- 
site to  that  of   the  original  field.     Representing  the  original 


280  MAGNETISM  [X,  §  197 

field  due  to  the  solenoid  by  H;,  and  the  reversed  polar  field  by 
Hp,  the  effective  magnetizing  field  operative  on  the  bar  is 

H  =  H<  -  Hp. 

Hence,  the  induction  density  in  the  bar1  is  given  by  the 
equation 

(22)  B=(Ht.-Hp)+47rl. 

The  resultant  value  of  the  magnetizing  field  strength  will  then 
be  less  than  H, ;  indeed  it  will  vary  from  point  to  point  along 
the  bar.  A  numerical  example  will  assist  in  making  this 
clear. 

Consider  a  bar  20  centimeters  long  and  2  square  centimeters 
in  cross-section  placed  in  a  magnetizing  coil  which  is  capable 
of  producing  a  field  of  50  gausses  (Fig.  134).  Let  us  assume 


FIG.  134. 

further  that  the  intensity  of  magnetization  of  the  iron,  due  to 
the  field  of  the  coil,  is  1000  units.  The  pole  strength  is,  from 
equation  (2), 

m=\a  =  2000. 

If  we  imagine  a  minute  longitudinal  crevasse  at  the  center 
of  the  bar  0,  a  unit  north  pole  placed  there  will  be  acted  on  by 
two  forces,  equal  in  magnitude  and  having  the  same  direction, 
which  is  opposite  to  that  of  H,.  One  force  is  due  to  the  re- 
pulsion of  N  and  the  other  to  the  attraction  of  S.  The  sum 
of  these  magnetic  forces  at  0  is  given  by  Coulomb's  law, 

40  dynes. 

J 


r2      r2         100 
For  a  permanent  bar  magnet  H<  =  0,  and  B  =  4  wl  — 


X,  §  197]  THE  MAGNETIC  CIRCUIT  281 

This  force  is  acting  in  opposition  to  the  field  due  to  the 
solenoid,  and  the  net  field  effective  on  the  bar  at  its  center  is 
10  units,  which  is  20%  of  the  original  field.  If  the  bar  is 
taken  twice  as  long,  that  is,  40  centimeters  long,  we  shall  have 

=10  dynes, 

so  that  the  net  field  strength  at  the  middle  of  the  bar  is  40 
units,  which  is  80  %  of  the  original  field.  Again,  if  we  con- 
sider a  bar  200  centimeters  long,  we  shall  have 

jpT==2^ooo_  =  04unit 

10000 

so  that  the  net  field  strength  at  0  is  49.6  units,  which  is  99.2  % 
of  the  original  value. 

It  is  clear,  therefore,  that,  in  general,  as  the  bar  increases 
in  length,  the  effect  of  the  poles  in  decreasing  the  magnetizing 
field  decreases;  but  the  effect  becomes  negligible  only  for  a 
very  long  bar.  For  a  bar  whose  length  is  large  as  compared 
to  its  cross-section,  it  is  clear  that  the  magnetic  induction  is 
greater  than  for  a  short  bar,  if  the  permeability  and  the  ex- 
ternal field  remain  the  same.  The  influence  of  the  ends 
becomes  negligible  only  when  the  ratio  of  length  to  diameter 
is  large,  say  from  200  to  400.  For  such  long  rods  the  effective 
magnetizing  force  is  practically  the  same  as  that  of  the  original 
field. 

Polar  demagnetization  is  aided  by  vibration  and  resisted  by 
coercivity.  (See  §  211.)  It  is  strongest  in  the  case  of  short 
bars.  These  will  almost  completely  demagnetize  themselves 
on  withdrawal  from  the  magnetizing  field. 

Short  bars  of  iron  used  by  themselves  are  not  useful  as  test 
pieces.  For  long  cylindrical  or  square  bars,  the  effect  of  the 
ends  can  be  determined l  approximately  and  applied  as  a  cor- 

1  Tables  of  demagnetizing  factors  for  bars  of  various  shapes  and  lengths 
are  found  in  the  larger  works  on  magnetism,  and  in  the  journals. 


282  MAGNETISM  [X,  §  197 

rection  factor.  For  ellipsoids  of  revolution  such  corrections 
can  be  precisely  calculated. 

Cylindrical  bars  sufficiently  long  to  render  the  end  effects 
negligible  are  neither  conveniently  made  nor  tested,  and  me- 
chanical difficulties  preclude  the  preparation  of  ellipsoidal 
specimens  except  in  well  equipped  standardizing  laboratories. 
Hence,  it  is  necessary  to  secure  the  condition  of  endlessness 
either  by  using  the  specimen  in  the  form  of  a  ring,  in  which 
case  there  are  no  poles,  or  to  approximate  this  condition  by 
clamping  the  bar  in  massive  yokes. 

198.  Magnetic  Leakage.     Iron  offers  an  easier  path  to  the 
flux  lines  than  air,  but  in  an  imperfect  magnetic  circuit  the 
flux  lines  do  not  all  follow  the  iron.     There  is  no  insulator 
for  magnetic  flux  and  therefore  it  cannot  be  confined  to  the 

conductor,  but  the  total 
flux  is  practically  constant 
through  every  chosen  cross- 
section  of  the  conductor. 

Consider  two  poles  of  a 
dynamo  (Fig.  135),  with  a 

FlG  135  total   flux   <£3.     A  part  of 

this,  <£b  is  in  a  position  to 

be  linked  with  the  wire  turns  of  the  armature,  and  a  part  <£2 
is  not  useful.  The  leakage  coefficient,  or  leakage  factor,  is 
the  ratio  of  the  total  to  the  useful  flux,  that  is 

(23)  fc  =  *§. 

9i 

199.  Laboratory   Exercise   XLV.     To   study   the   magnetic 
leakage  about  a  magnetic  circuit. 

APPARATUS.  Electromagnet  with  armature,  or  the  field 
coils  of  a  dynamo.  Reversing  switch,  ammeter  and  source  of 
current,  search  coils,  and  ballistic  galvanometer. 


X,  §  199] 


THE   MAGNETIC  CIRCUIT 


283 


PROCEDURE.  (1)  Pass  current  through  the  field  coils  of 
such  value  that  the  desired  magnetic  flux  is  assured.  Place 
a  search  coil  at  A  (Fig.  136),  and  connect  its  terminals  to  the 
ballistic  galvanometer.  Eeverse  the  current  and  read  the 
galvanometer  throw.  Record  cur- 
rent, coil  position,  and  deflection. 
Take  several  readings. 

(2)  Move  the  search  coil  to  B  and 
repeat  (1). 

(3)  Again  repeat  (1)  with  the  coil 
at  C. 

(4)  Calculate  values  for  the  leakage 
coefficient  for  position  C. 

(5)  If  actual   values    of    the   flux 
are  required,  the  ballistic  galvanom- 
eter may  be  calibrated  by  the  method  given  in  §  151. 

(6)  Separate  the  armature  C  from  the  pole  pieces  by  a  single 
thickness  of  paper  at  aa',  and  repeat  (1).     Insert  at  aa'  cal- 
ipered  pieces  of  thin  brass,  and  repeat  (1),  studying  the  change 
in  k  with  length  of  air  gap.     For  comparison  of  deflections  the 
same  value  of  the  exciting  current  should  be  used  throughout. 


CHAPTER   XI 


THE   EARTH'S   MAGNETISM 

200.  The  Magnetic  Elements.  Near  the  close  of  the  six- 
teenth century  the  investigations  of  Gilbert  established  the 

magnetic  nature  of  the 
earth.  It  is  a  weak  mag- 
net for  which  the  value  of 
the  intensity  of  magnetiza- 
tion is  about  0.08.  The 
phenomena  observed  on  the 
earth's  surface  are  prac- 
tically those  which  would 
be  present  if  a  bar  magnet, 
short  as  compared  to  the 
earth's  radius,  were  located 
within  the  earth,  as  shown 
in  Fig.  137.  The  geo- 
graphic axis  is  represented 
by  N'S't  the  equator  by 
EE',  and  the  magnetic 

axis  by  NS.  The  dotted  lines  show  the  direction  of  the  field, 
which  is  the  direction  in  which  a  north-seeking  pole  will 
point. 

From  the  law  of  magnetic  pole  attraction  we  learn  that 
unlike  poles  attract;  hence,  the  magnetic  pole  of  the  earth 
which  is  geographically  north  is  opposite  in  kind  from  the 
pole  of  the  compass  needle  which  seeks  that  direction. 
Confusion  on  this  point  will  be  avoided  if  it  is  recognized 

284 


XI,  §  200]     EXPERIMENTAL  DETERMINATIONS         285 

that  the  north  magnetic  pole  takes  its  name  from  being 
near  the  geographic  north  pole,  while  the  north-seeking  pole 
owes  its  name  to  the  fact  that  it  points  towards  the  north. 
The  lines  of  force  of  the  earth's  field  must  be  considered 
as  having  a  direction  towards  the  north. 

The  magnetic  axis  departs  about  15°  from  the  direction  of 
the  axis  of  rotation  of  the  earth.  Its  points  of  intersection 
with  the  earth's  surface  are  called  the  magnetic  poles.  These 
positions  will  be  found  marked  on  any  of  the  larger  maps  of 
the  world.  The  north  magnetic  pole  is  approximately  on  the 
meridian  through  Omaha,  and  about  500  miles  north  of  Hud- 
son Bay,  just  above  latitude  70°  N.  The  south  magnetic 
pole  is  on  the  meridian  of  Eastern  Australia,  about  lati- 
tude 73°  S. 

A  magnetic  meridian  is  a  vertical  plane  which  passes  through 
the  magnetic  axis  of  a  freely  suspended  magnetic  needle  which 
is  in  equilibrium. 

In  order  to  specify  precisely  the  earth's  magnetic  field  at 
any  point,  three  elements  are  usually  given  as  follows : 

(1)  The  inclination  or  angle  of  dip,  which  is  the  angle  be- 
tween the  magnetic  axis  of  a  freely  suspended  magnetic  needle 
and  a  horizontal  line  through  its  center. 

(2)  The  declination,  which  is  the  angle  between  the  mag- 
netic and  geographic  meridians.     It  is  the  angle  of  departure 
of  the  needle  from  true  north. 

(3)  The  intensity  of  the  field,  which  is  the  force 
in  dynes  which  acts  on  a  unit  pole  placed  in  the 
field. 

In  measuring  the  intensity  it  is  usually  most 
convenient  to  determine  the  horizontal  component 
of  the  earth's  field,  represented  by  the  symbol  H9 
Fig.  138,  and  also  the  angle  of  dip  a,  whence  the  total  force  F 
is  found  from  the  relation 
(1)  H=Fcosa. 


286  THE  EARTH'S   MAGNETISM          [XI,  §  200 

The  vertical  component  may  be  found  from 
(2)  F=Fsina. 

201.  Magnetic  Surveys.  For  many  years  accurate  magnetic 
surveys  have  been  made  over  all  parts  of  the  earth's  surface 
by  the  scientific  bureaus  of  various  countries,  and  detailed 
records  and  charts  have  heen  prepared  which  show  the  values 
of  the  magnetic  elements  and  their  variations  for  any  given 
position.  These  studies  show  daily,  annual,  and  eleven-year 
period  variations,  all  of  which  are  due  to  the  sun ;  a  possible 
variation  due  to  the  moon ;  secular  or  long-period  variations 
of  unknown  cause,  which  extend  over  centuries ;  and  certain 
irregular  and  occasional  variations  connected  with  magnetic 
storms  and  auroral  displays. 

At  any  given  point,  however,  the  earth's  field  is,  for  short 
periods,  practically  constant.  Accurate  determinations  of  the 
magnetic  elements  are  of  primary  importance  for  the  navigator 
and  for  the  surveyor.  Formerly  they  were  also  essential  in 
the  electric  laboratory  in  connection  with  absolute  measure- 
ments. The  horizontal  component  of  the  earth's  field  can  be 
determined  easily  in  absolute  measure,  and  it  is  a  valuable 
secondary  standard  in  such  problems  as  current  measurements 
with  the  tangent  galvanometer,  or  ballistic  galvanometer  cali- 
brations with  the  earth  inductor. 

At  present,  however,  such  methods  are  obsolete  for  the  pur- 
pose of  electric  measurement  on  account  of  the  general  use 
of  iron  and  steel  in  buildings,  and  on  account  of  the  stray 
fields  from  electric  machinery  and  direct  current  distribut- 
ing circuits.  Nevertheless,  a  brief  treatment  of  a  few  mag- 
netic measurements  will  be  given  because  of  their  intrinsic 
interest. 

The  student  should  read  the  chapters  on  the  earth's  mag- 
netism in  the  larger  textbooks  of  physics,  and  study  carefully 
the  significance  of  the  lines  drawn  on  the  magnetic  charts. 


XI,  §  202]     EXPERIMENTAL  DETERMINATIONS         287 

202.  Laboratory  Exercise  XL VI.  To  find  the  inclination 
with  the  dip  circle. 

APPARATUS.     Dip  circle  and  accessories. 

The  dip  circle  (Fig.  139)  consists  essentially  of  a  magnetic 
needle  symmetrically  mounted  on  a  steel  staff  whose  polished 


FIG.  139. 

cylindrical  ends  roll  on  agate  plates.  If  (a)  the  needle  swings 
freely  in  a  vertical  plane  coincident  with  the  magnetic  merid- 
ian, (b)  its  center  of  gravity  lies  exactly  in  the  axis  of  rota- 
tion, and  (c)  its  geometric  axis  coincides  with  its  magnetic 
axis,  then,  if  there  is  no  friction,  the  needle  will  take  a  posi- 
tion with  its  axis  strictly  parallel  to  the  lines  of  force  of  the 


288  THE  EARTH'S  MAGNETISM          [XI,  §  202 

earth's  field,  and  the  angle  between  the  axis  and  the  horizontal 
as  read  on  the  vertical  circle  gives  the  dip. 

Three  sources  of  error  must  be  taken  into  account  when  using  the 
instrument. 

1.  The  polished  cylindrical  ends  of  the  supporting  staff  roll  on  the 
agate  plates,  which  prevents  perfect  coincidence  of  the  axis  of  rotation 
with  the  center  of  the  vertical  circle.    The  mean  of  readings  taken  at  both 
ends  of  the  needle  will  be  free  from  this  error. 

2.  The  center  of  gravity  of  the  needle  may  not  coincide  with  the  axis 
of  rotation,  in  which  case  the  observed  angle  will  be  increased  or  dimin- 
ished according  to  the  relative  position  of  the  center  of  gravity  and  the 
axis.    The  mean  of  readings  taken  with  the  polarity  of  the  needle  reversed 
will  be  free  from  this  error. 

3.  The  geometric  and  magnetic  axes  may  not  coincide.     The  mean 
of  readings  taken  with  the  opposite  ends  of  the  supporting  staff  toward 
the  observer  will  be  free  from  this  error. 

PROCEDURE.  (1)  Level  the  instrument  carefully,  so  that 
for  any  position  of  the  horizontal  circle  the  bubble  seeks  the 
middle  of  the  tube.  Release  the  arrestment,  thus  lowering 
the  needle  on  to  the  agate  plates,  and  see  that  it  swings  freely. 

(2)  Rotate  the  instrument  about  a  vertical  axis  until  the 
needle  assumes  a  vertical  position.     The  needle  now  lies  in 
a  plane  at  right  angles  to  the  magnetic  meridian.     Rotate  the 
instrument  again  in  azimuth  through  90°  and  clamp  it.     The 
needle  now  lies  in  the  magnetic  meridian. 

(3)  Read  both  ends  of  the  needle.     Just  before  taking  the 
readings  it  is  well  to  tap  the  base  of  the  instrument  gently 
with  the  finger  in  order  to  overcome  any  static  friction  be- 
tween the  staff  and  the  agate  plates. 

(4)  Rotate  the  instrument  in  azimuth  through  180°  and  read 
both  ends  as  before. 

(5)  Tabulate  all  data  and  compute  the  mean  value  of  the 
angle  of  dip. 

203.  The  Horizontal  Component  of  the  Earth's  Field  in 
Absolute  Measure.  This  method  involves  the  use  of  a  mag- 


XI,  §  203]     EXPERIMENTAL  DETERMINATIONS         289 

netometer,  which  consists  essentially  of  a  magnetic  needle  sus- 
pended by  a  light  fiber  and  capable  of  rotating  freely  in  a 
horizontal  plane.  A  small  mirror  is  attached  to  the  needle, 
and  the  deflections  can  be  observed  by  the  use  of  a  telescope 
and  scale.  A  bar  magnet  of  moment  M  is  selected.  By  means 
of  magnetometer  deflections  and  the  equations  derived  below, 
values  will  be  found  for  the  product  MHsmd.  for  the  quotient 
M/H,  where  H  represents  the  horizontal  component  of  the 
earth's  field.  The  quantity  M  may  be  eliminated  from  these 
expressions,  and  the  value  of  H  may  be  found  in  terms  of 
deflections  and  distances. 

I.    To  find  the  value  of  MIL 

We  shall  first  develop  a  simple  relation  between  the  quantity 
MH  and  the  periodic  time  of  vibration  of  the  suspended 
magnet.  The  magnet  used  will  be  a  cylindrical  bar  about  ten 
centimeters  long  and  a  few  millimeters  in  diameter.  This  bar, 
represented  by  ns  in  Fig.  140,  is  hung  in  a  suitable  stirrup  by 


Hm 


FIG.  140. 

a  nearly  torsionless  fiber  attached  at  o.  When  it  has  been 
deflected  through  a  small  angle  a  and  then  released,  it  will 
vibrate  in  a  horizontal  plane  until  its  energy  is  dissipated  by 
friction  against  the  air.  The  magnetic  meridian  is  represented 
by  NS,  the  pole  strength  of  the  suspended  magnet  by  m,  and 
the  distance  between  the  poles  by  I.  When  a  pole  of  strength 
m  is  placed  in  a  field  of  strength  H,  it  will  be  acted  on  by 
u 


290 


THE   EARTH'S   MAGNETISM 


[XI,  §  203 


a  force  of  Hm  dynes.     The  magnet  when  deflected  through 
some  angle  a,  tends  to  regain  its  equilibrium  position  due  to 


Jim 


Hm 

FIG.  140  (repeated). 

the  two  forces  acting  on  its  two  poles.     The  moment  of  this 
force  couple,  the  so-called  restoring  couple,  is  given  by 

(3)  L  =  Hml  sin  a. 

If  it  is  assumed  that  a  is  so  small  that  the  sine  does  not  differ 
sensibly  from  the  angle  expressed  in  radians,1  then  we  have 

(4)  L  =  Hmla, 
which  may  be  written  in  the  form 

(5)  L  =  HMa. 

From   (5)  it  is  seen  that   the  restoring  moment,  or   torque, 
is  directly  proportional  to  the  angular  displacement;   hence, 

1  For  the  sake  of  definite  comparison,  the  following  table  is  given  : 


ANGLE  IN  DEGREES 

ANGLE  IN  KADIANS 

SINE 

1° 

0.01745 

0.017453 

2 

0.03490 

0.034906 

5 

0.08720 

0.087165 

10 

0.17360 

0.17453 

For  an  angle  of  10°,  the  error  due  to  the  substitution  of  the  angle  for  the 
sine  is  slightly  more  than  one  half  of  one  per  cent.  The  student  should  verify 
this  and  he  should  construct  other  examples  from  a  trigonometric  table. 


XI,  §  203]      EXPERIMENTAL   DETERMINATIONS         291 


the  suspended  magnet  is  vibrating  with  simple  harmonic 
motion.  The  periodic  time  of  such  motion  is  given  by  the 
formula 
(6) 

where  K  is  the  moment  of  inertia  of  the  bar.     Substituting  in 
(6)  the  value  of  the  torque  from  (5),  we  have 

~Ka 


whence 

(7) 
or 

(8) 


T2 


II.    To  find 'the  value  of  M/H. 

Two  different  arrangements  of  the  deflecting  magnet  NS, 
with  respect  to  the  needle  of  the  magnetometer  0,  will  each 
give  a  value  of  M/H  in  terms  of  observed  deflections.  These 
positions  are  shown  in  Figs.  141  and  142,  respectively.  They 

m is 


\N' 


N\         \8 


\S' 


FIG.  141. 


8' 
FIG.  142. 


are  known  as  the  tangent  position  and  the  broadside  position. 
The  latter  is  the  more  sensitive  and  the  theory  will  be  developed 
for  that  case. 


292 


THE  EARTH'S  MAGNETISM 


[XI,  §  203 


In  Fig.  143,  NS  represents  the  deflecting  magnet  which 
has  a  pole  strength  m  and  an  interpolar  distance  2  I.  The 
magnetometer  needle  is  represented  by  ns,  and  it  has  a 
pole  strength  m'  and  a  length  I'.  The  deflecting  magnet  is 

placed  with  its  axis  at 
right  angles  to  the  mag- 
netic meridian  WS'. 

Let  us  assume  that  the 
length  of  ns  is  small  as 
compared  with  the  dis- 
tance r.  Then  we  may, 
without  appreciable  error, 
consider  the  triangle  NsS 
as  an  isosceles  triangle 
with  an  altitude  r. 

The  needle  ns,  originally 
in  the  magnetic  meridian, 
is  deflected  through  an 
angle  a  by  the  presence  of 
the  deflecting  magnet.  The  forces  acting  on  s  are  given  by 
the  equations  mm'  f  _mm' 

/i-- ^n  J^-w" 

The  resultant  of  these  two  forces  tends  to  deflect  the  south 
pole  to  the  west,  and  its  value  is  given  by  the  equation 

2mm'l 


F  =2 


mm 


cos  <  = 


since  cos  <£  =  l/d.     The  magnetic  moment  of  the  deflecting 
magnet  is,  by  definition,  M  —  2  ml;  substituting  this  in  (9)  gives 

(10)  F»  =  Mm'. 

d* 

The  moment  of  the  deflecting  couple  acting  on  the  needle  is 


FJ 


cos  a. 


XI,  §  203]     EXPERIMENTAL   DETERMINATIONS         293 

The  moment  of  the  restoring  couple  due  to  the  earth's  field  is 
Hm'l1  sin  a.  Equating  these  moments,  we  have 

^^-  cos  a  =  Hm'l'  sin  a, 
cP 

whence 

(11)  IT  tan  a  =  —  =  Md-\ 

ds 

The  desired  value  of  M/H  could  be  found  from  equation  (11) 
provided  the  magnetic  length  of  N8  were  known.  Since  this 
is  not  easily  determined,  it  is  necessary  to  transform  (11),  and 
to  arrange  the  experiment  so  that  this  length  may  be  eliminated 
from  the  equations. 

Substituting  for  d  in  equation  (11)  its  value  (r2  +  Z2)^,  and 
expanding,  we  have 

(12)  H  tan  a  =  M[r~*  -  f  rt2  +  »•]. 

Since  r  is  large,  negative  powers  above  the  fifth  may  be 
neglected.  Multiplying  both  sides  of,  (12)  by  r5,  we  have 

(13)  Hr6  tan  a  =  M [r2  -  f  Z2]. 

Repeating  the  procedure  just  described,  but  with  a  different 
distance  i\  and  a  corresponding  angle  a^  we  may  write 

(14)  Hr*  tan  a,  =  M[rf  -  f  Z2] . 

Eliminating  3  MP/2  between  equations  (13)  and  (14),  we 
obtain  the  equation 

(15)  H[r*  tan  a  -  r^  tan  oj  =  M  [r2  -  r^], 
whence 

d 6")  ^=  r6  tan  «  —  r^  tan  o^ 

H~  r2-^2 

If  M  remains  constant  throughout  the  experiment,  the  value 
of  H  may  be  calculated  from  the  values  of  MH  and  M/H  ob- 
tained above,  by  means  of  the  identity 


H 


294  THE  EARTH'S  MAGNETISM          [XI,  §  204 

204.  Laboratory  Exercise  XL VII.  To  determine  H  by  the 
magnetometer  method. 

APPARATUS.  Magnetometer  and  accessories,  stop  watch, 
meter  scale,  micrometer  calipers,  and  compass. 

PROCEDURE.  (1)  Hang  the  deflecting  magnet  in  a  stirrup, 
cover  it  with  a  bell  jar,  and  observe  the  time  of  vibration  at 
the  position  where  H  is  to  be  determined.  The  angle  of  swing 
should  not  exceed  5°,  and  at  least  five  determinations  should 
be  made  of  the  time  for  25  swings,  the  mean  time  of  one 
vibration  being  then  calculated. 

Throughout  the  experiment  no  magnetic  material  other  than 
the  magnet  in  use  should  be  about  the  table,  or  on  tjie  person 
of  the  observer. 

(2)  Measure  the  length  and  mean  diameter  of  the  magnet, 
and  find  its  mass.  From  these  data  compute  its  moment  of 
inertia  by  means  of  the  formula 


where  d  and  L  are  the  diameter  and  length  of  the  magnet, 
respectively. 

Calculate  from  equation  (8)  the  value  of  MH. 

(3)  Set  the  table  of  the  magnetometer  with  its  axis  in  the 
magnetic  meridian  as  determined  by  the  compass,  and  level 
the  instrument  so  that  the  needle  swings  freely.  Set  the 
telescope  and  scale  at  a  distance  of  one  meter  in  front  of  the 
magnetometer,  being  careful  to  have  the  scale  parallel  to  the 
magnetometer  table. 

With  the  deflecting  magnet  removed  to  a  distant  part  of  the 
room,  take  the  zero  reading  on  the  scale.  Place  the  magnet 
symmetrically  in  its  support  at  a  distance  r  from  the  needle 
and  read  the  deflection  a'.  Turn  the  magnet  end  for  end  and 
again  read  the  deflection  a".  The  mean  of  these  readings,  a, 


XI,  §  205]     EXPERIMENTAL  DETERMINATIONS        295 

will  be  free  from  error  due  to  lack  of  symmetry  in  the  distri- 
bution of  the  magnetism  along  the  deflecting  magnet. 

Place  the  magnet  on  the  other  end  of  the  table  at  the  cor- 
responding distance  and  take  deflections  with  the  north  pole 
toward  the  east,  and  again  toward  the  west,  as  before.  From 
these  four  readings  the  mean  deflection  for  the  distance  r  will 
be  found. 

(4)  Repeat  (3)  for  another  position  at  a  distance  TI  from  the 
needle,  for  which  the  deflection  «j  will  be  observed.     It  is  well 
to  take  readings  also  for  two  other  values  of  r,  thus  insuring 
two  independent  determinations  of  M/H. 

(5)  Substitute  these  data  in  equation  (16). 

(6)  With  the  values  found  for  MH  and  M/H,  calculate  from 
the  identity  (17)  the  value  of  H.     Express  the  result  in  appro- 
priate units. 

Carelessness  in  handling  the  deflecting  magnet,  such  as  striking, 
jarring,  or  dropping  it,  may  change  the  value  of  its  magnetic  moment. 
Only  in  case  M  remains  constant  throughout  the  progress  of  the  experi- 
ment, is  its  elimination  in  equation  (17)  legitimate. 

Tabulate  all  data,  arranging  the  table  to  show  the  number  of  the  de- 
flecting magnet  used,  its  mass,  dimensions  and  moment  of  inertia,  the 
observations  for  and  the  final  value  of  its  time  of  vibration,  the  separate 
and  mean  values  of  r  and  a,  and  the  computed  values  of  M H,  M/H,  and  H. 

Explain  why  a  suspended  magnetic  needle  experiences  a  rotation  only 
and  not  a  motion  of  translation. 

205.  Laboratory  Exercise  XL VIII.  To  determine  H  with  the 
standard  tangent  galvanometer  and  copper  voltameter. 

APPARATUS.  Standard  tangent  galvanometer,  one  or  two 
storage  cells,  control  rheostat,  reversing  switch,  and  two  copper 
voltameters  with  accessories. 

From  equations  (17),  §  107,  and  (24),  §  108,  it  is  seen  that  H 
may  be  expressed  in  terms  of  a  current  strength,  the  constants 
of  a  tangent  galvanometer,  and  the  tangent  of  the  deflection 
angle.  The  current  can  be  accurately  measured  with  a  copper 
voltameter. 


296 


THE   EARTH'S  MAGNETISM          [XI,  §  205 


PROCEDURE.  (1)  Arrange  the  circuit  as  in  Fig.  144.  The 
current  from  the  storage  cells  may  be  adjusted  to  a  suitable 
value  by  means  of  the  rheostat  It,  and  it  may  be  controlled 
during  the  progress  of  the  experiment  if  there  is  any  tendency 
toward  fluctuation. 

The  reversing  switch  S  reverses  the  current  through  the 
galvanometer  but  not  through  the  voltameters  W.  Two 


FIG.  144. 

voltameters  may  be  connected  in  series,  one  serving  as  a  check 
on  the  other.  The  deposits  on  the  cathodes  of  each  should 
be  the  same.  In  each  cell  the  outside  plates  will  be  anodes, 
the  cathode  being  held  midway  between  them. 

(2)  Clean  the  surfaces  of  the  anodes  with  sandpaper,  rinse 
them  in  clean  water,  and  place  them  in  the  voltameters.     The 
cathode  plates  will  be  carefully  cleaned  with  sandpaper,  laying 
them  meanwhile  on  a  piece  of  clean  paper  and  avoiding  touch- 
ing them  with  the  fingers.     When  clean  and  bright,  rinse  them 
in  clean  water,  then  in  alcohol,  and  dry  them  by  twirling  them 
for  a  few  seconds  in  the  air.     They  should  then  be  weighed 
with  great  care  and  folded  in  a  sheet  of  clean  paper  until 
ready  to  use. 

(3)  Insert  a  pair  of  trial  cathodes  and  adjust  the  resistance 
R  until  the  galvanometer  shows  a  deflection  of  about  45°. 


XI,  §  206]     EXPERIMENTAL  DETERMINATIONS         297 

Note  whether  the  galvanometer  needle  deflects  equally  on 
both  sides  of  zero,  and  whether  the  readings  of  opposite  ends 
of  the  needle  are  nearly  the  same.  If  not,  adjustments  of  the 
leveling  screws  and  torsion  head  are  necessary. 

When  the  galvanometer  and  current  are  properly  adjusted, 
open  the  switch  $,  replace  the  trial  plates  by  the  weighed 
cathodes  and  close  the  switch,  noting  the  exact  time.  The 
cathode  plates  should  be  fully  immersed. 

(4)  Let  the  current  pass  for  30  or  60  minutes.     Observe  the 
galvanometer  deflection,  reading  both  ends  of  the  needle  every 
two  minutes.      Immediately  after   each   reading,  throw  over 
the  switch  &.     Record  in  a  table   the  readings  of  time  and 
deflections. 

The  current  should  be  kept  constant  by  adjusting  E  if 
necessary. 

(5)  At  the  end  of  the  period  break  the  circuit,  reading  the 
exact  time.     Quickly  remove  the  cathodes  and  plunge  them  into 
a  bath  of  water  made  slightly  acid  with  H2S04.     Rinse  them 
in  water,  then  in  alcohol,  and  dry  them  as  before.      Again 
weigh  them  carefully. 

(6)  Calculate  the  current  strength  from  equation  (45),  §  113. 
The  electro-chemical  equivalent  of  copper  varies  slightly  with 
the  temperature  and  current  density,  but  the  value  0.0003294 
grams  per  coulomb  may  be  used  here  with  safety. 

(7)  Calculate  the  value  ef  H  from  equation  (24),  §  108. 

The  watch  used  should  be  compared  before  and  after  the  experiment 
with  the  standard  clock,  and  a  correction  for  its  rate  should  be  applied  if 
necessary. 

The  electrolyte  should  be  made  from  pure  CuSO4  and  water  with  a 
density  of  about  1.18.  One  per  cent  of  H2SO4  should  be  added. 

206.   Laboratory  Exercise  XLIX.     To  compare  values  of  H. 
APPARATUS.     Small  magnet  with  support  and  bell  jar,  and 
watch. 


298  THE   EARTH'S   MAGNETISM          [XI,  §  206 

When  the  value  of  H  is  known  at  one  station  the  value  at  any  other 
station  is  readily  found  by  the  use  of  equation  (8).  Let  H  and  IF  be  the 
values  at  two  stations,  and  let  T  and  T  be  the  corresponding  periods  of 
vibration  of  the  suspended  magnet.  For  the  same  magnet  M  and  K  will 
be  constant,  and  we  shall  have 


(18)  r=2ir\--.        T  = 

' 


Squaring  both  of  these  equations  and  dividing  the  first  of  them  by  the 
second,  we  have 

(19>  f=-p- 

PROCEDURE.  (1)  Support  the  vibrating  magnetic  needle 
under  the  bell  jar  at  the  station  where  H  is  known,  and 
take  the  time  of  25  or  50  vibrations,  keeping  the  angle  small. 
Repeat  several  times  and  calculate  the  mean  value  of  T. 

(2)  Remove  the  apparatus  to  the  station  where  H'  is  to  be 
found,  repeat  the  observations  for  the  time  of  vibration,  and 
find  the  mean  value  for  T'. 

(3)  Calculate  from  equation  (19)  the  value  for  H'  at   the 
second  position. 

Note  for  each  position  the  location  of  the  needle  with  respect 
to  any  structural  iron  work,  or  other  fixed  masses  of  magnetic 
material,  and  explain  their  probable  effect  on  the  period  of  the 
needle. 

In  this  way  a  magnetic  survey,  of  any  locality  is  readily 
carried  out. 

The  method  is  analogous  to  that  used  for  comparing  values  of  the 
acceleration  due  to  gravity,  which  involves  a  similar  treatment  of  the 
formula  for  the  simple  pendulum, 

T  =  2 

207.  The  Earth  Inductor.  A  device  frequently  used  for 
producing  a  known  charge,  and  for  measuring  indirectly  the 
horizontal  or  vertical  components  of  the  earth's  field,  is 


XI,  §  207]     EXPERIMENTAL   DETERMINATIONS         299 


the  so-called  earth  inductor  shown  in  section  in  Fig.  145.     It 

consists  of  a  noii-inagnetic  frame  of  round  or  square  section, 

on   which  is  wound  a  coil  of  wire   of   a 

known  number  of  turns  S,  and  of  effec- 

tive radius  r.     This  coil  is   so  mounted 

that  it  may  be  rotated  about  either  a  hori- 

zontal or  a  vertical  axis,  and  it  is  pro- 

vided with  a  spring  release  and  a  stop, 

so  that  it  may  be  rapidly  turned  through 

an  angle  of  180°.     If  the  horizontal 


FIG.  145. 


ponent  of  the  earth's  field  is  accurately  known,  and  if  the  coil 
is  rotated  about  a  vertical  axis,  the  quantity  Q  induced  in  the 
circuit  of  the  coil  in  one  half  a  rotation  may  be  computed  by 
the  formula 

2  SHA 


(20) 


R 


where  A  is  the  area  of  the  coil  and  R  is  the  resistance  of  the 
circuit.  This  quantity  may  be  used  for  the  calibration  of  a 
ballistic  galvanometer.  On  the  other  hand,  equation  (20)  may 
be  used  for  the  determination  of  H  itself. 

If  deflections  are  read  on  a  ballistic  galvanometer  for  both 
vertical  and  horizontal  positions  of  the  plane  of  the  coil,  the 
horizontal  and  vertical  components  of  the  earth's  field,  respec- 
tively, are  linked  with  the  wire  turns,  and  the  angle  of  dip  may 
be  determined  even  though  values  of  V  and  H  are  not  known. 
If  c?!  denotes  the  observed  deflection  when  the  coil  is  horizon- 
tal, and  c?2  the  deflection  when  the  coil  is  vertical,  the  corre- 
sponding charges  Ql  and  Q2  are  given  by  the  equations 


(21) 


2SVA 

5  — 
£1 


(22) 


2  SHA 


300  THE  EARTH'S  MAGNETISM  [XI,  §  207 

Dividing  (21)  by  (22),  we  obtain  the  relation 

Z=:i 

H     di 

From  equations  (1)  and  (2),  we  have 


-  =  tan«, 


whence 
(23) 


The  number  of  turns  and  the  dimensions  of  the  coil  must  be 
accurately  known,  and  the  horizontal  and  vertical  positions 
must  be  carefully  determined  with  a  spirit  level.  The  values 
of  V  and  H  are  subject  to  large  variations  due  to  local  con- 
ditions, but  the  conditions  may  be  regarded  as  remaining  con- 
stant during  the  time  required  for  these  observations. 


CHAPTER  XII 

MAGNETIC   TESTING 

PART  I.     MAGNETIZATION  CURVES  —  HYSTERESIS 

208.  The  Importance  and  the  Nature  of  Magnetic  Tests.1 

The  industrial  importance  of  iron  and  steel  from  the  magnetic 
viewpoint  is  very  great.  According  to  the  purpose  for  which 
it  is  to  be  used,  it  must  possess  high  permeability,  or  low  dis- 
sipation of  energy  in  the  process  of  magnetization,  or  the 
capacity  of  retaining  a  large  percentage  of  its  induced  ma.g- 
netism. 

Hence,  continued  and  systematic  testing  of  the  magnetic 
qualities  of  iron  and  steel  is  necessary  for  the  producer  of  the 
material,  the  designer,  and  the  manufacturer  of  electromagnetic 
machinery. 

The  producer  must  maintain  a  close  control  over  his  output 
in  order  to  take  advantage  of  the  great  variation  in  magnetic 
quality  arising  from  slight  variations  in  the  composition  and 
treatment  of  the  materials. 

The  manufacturer  must  compare  the  predetermined  efficiency 
of  the  design  with  the  actual  performance  of  the  completed 
apparatus.  In  recent  years  the  quality  of  the  iron  and  steel 
produced  for  magnetic  purposes  has  been  greatly  improved, 
and  to-day  more  than  ever  before,  the  results  of  careful  tests 
are  being  studied. 

There  are,  in  general,  three  kinds  of  tests  to  which  iron  and 
steel  are  subjected. 

1  A  comprehensive  treatment  of  the  various  magnetic  tests,  together  with 
the  modern  methods,  typical  data,  and  results  will  be  found  in  Circular  of 
U.  S.  BUREAU  OF  STANDARDS,  No.  17,  Magnetic  Testing. 

301 


302 


MAGNETIC  TESTING 


[XII,  §  208 


I.  B-H  curves.  A  test  to  determine  values  of  the  flux 
density  in  the  material  for  a  given  set  of  values  of  the  mag- 
netizing field.  From  this  the  permeability  may  be  computed. 


15 


10 


B-H  Curve 


H  Cur 


H 


2     4 


8    10 


gauss 


30 


Fm    146. 


II.  Hysteresis   and    core-loss.     A   test   to   determine   the 
energy  expended  in  carrying  the  sample  through  a  complete 
magnetic  cycle.     Work  is  done  also   in  setting  up  eddy  cur- 
rents within  the  metal.     The  energy  consumed  per  second  due 
to  both  of  these  causes  is  called  the  core-loss. 

III.  Residual    Magnetism.       A     test    to    determine    the 
amount  of  magnetism  retained  after  the  magnetizing  field  has 
been  withdrawn.     This    may  also  include   the   determination 
of  the  tenacity  with  which  the  magnetism  is  held. 


XII,  §  209] 


MAGNETIZATION  CURVES 


303 


209.  Units  of  H  and  B.  The  designer  of  electrical  machin- 
ery must  know  the  flux  density  which  a  given  magnetizing  field 
will  establish  in  the  material  used.  He  must  also  know  the 
value  of  the  permeability,  and  how  it  varies  with  B  and  H.  A 


& 

16 
15 

14 

11 
10 
9 
S 
7 
6 
5 

4 

3 

2 

1 

H   i 

B  < 

; 

•\ 

s~~ 

— 

\l 

\ 

yf 

\ 

I 

/ 

\ 

B 

•K 

lun 

e 

/ 

\ 

\ 

I/ 

\ 

\ 

/ 

H 

^N 

sCw 

-ve 

\ 

I! 

x 

v^ 

\ 

I 

v>>>> 

^ 

^ 

\ 

I 

, 

^ 

^s 

^-. 

I 

X 

1 

1      2      4       6       S      10    12     14     16    18     20     22     24     26    28     30      gauss 
)      1       234       S      6       7      S      9     10    11     12     13     14     15      kg. 
FIG.  147. 

typical  B-H  curve  for  cast  steel  is  shown  in  I,  Fig.  146.  In  Fig. 
147  the  B-/x  and  H-/A  curves  are  shown  for  the  same  material. 

The  magnetizing  field  strength  H  is  commonly  expressed  in 
C.  G-.  S.  lines  (or  maxwells)  per  square  centimeter,  that  is  in 
gausses ;  or  in  gilberts  per  centimeter ;  or  in  ampere-turns 
per  centimeter  or  per  inch. 

The  induction  density  B  is  expressed  in  gausses,  or  in  kilo- 


304 


MAGNETIC  TESTING 


[XII,  §  209 


gausses,  or  in  lines  per  square  inch.  Corresponding  values  of 
the  important  magnetic  quantities  (§§  187-193)  for  the  sample 
of  cast  steel  mentioned  above,  are  shown  in  the  following 
table,  with  H  and  B  in  gausses. 


H 

B 

I 

k 

/"• 

1.0 

650 

51.7 

51.7 

650 

4.0 

5300 

421.7 

105.4 

1325 

10.0 

10300 

819.3 

81.9 

1030 

20.0 

13100 

1041.4 

52.1 

655 

210.  B-H  Curves.  The  upper  curve  in  Fig.  146  and  the 
upper  ones  in  Fig.  148  show  three  distinct  stages  of  the  mag- 
netizing process.  At  first,  for  low  values  of  H,  the  induction 
increases  slowly.  Next,  the  induction  rises  rapidly  with  large 
changes  for  small  increments  of  H,  and  finally,  after  reaching 
the  knee  of  the  curve,  further  increase  is  slow  even  for  large 
increments  of  the  magnetizing  force. 

The  value  of  B  can  always  be  increased  by  increasing  H, 
but  a  limit  is  soon  reached  above  which  it  is  not  practicable 
to  go.  At  this  stage  the  iron  is  said  to  be  approaching  satura- 
tion. For  wrought  iron  and  cast  steel  this  limit  is  approxi- 
mately reached  at  15,000  to  17,000  gausses.  For  cast  iron  the 
saturation  point  is  at  about  10,000  gausses.  Curves  for  differ- 
ent sorts  of  iron  and  steel  are  shown  in  Fig.  148. 

The  induction  density  depends  somewhat  upon  the  initial 
state  of  the  sample,  and  upon  its  previous  magnetic  history, 
as  well  as  upon  the  rapidity  and  mode  of  change  from  one 
value  of  the  magnetizing  field  to  another.  The  discussion  of 
these  matters  will  be  resumed  in  the  following  articles.  The 
permeability  is  diminished  by  mechanical  treatment,  such  as 
rolling  or  hammering. 

The  testing  of  iron  at  low  values  of  the  induction  density 
has  become  important  in  recent  years  because  of  the  increas- 


XII,  §  210] 


MAGNETIZATION   CURVES 


305 


ing  use  of  relays  for  feeble  currents.  In  fields  where  H  is  less 
than  one  gauss,  the  induction  density  curve  is  essentially  a 
straight  line  starting  with  a  finite  inclination  to  the  H  axis. 
With  alternating  currents  of  high  frequencies,  that  is,  up- 
wards of  100,000  cycles  per  second,  the  permeability  probably 


15 


io\ 


7 


7 


Wrought  Iron 


Gad  Steel 


Cast  Iron^. 


H 


0      g      4     -ff      8     10 


gauss  20 

FIG.  148. 


SO 


does  not  differ  greatly  from  that  given  by  the  normal  induc- 
tion curve,  but  it  is  not  significant,  and  it  is  difficult  to  meas- 
ure because  of  the  skin-effect  due  to  the  rapid  alternations. 
The  induction  near  the  center  of  the  sample  is  exceedingly 
small,  because  before  the  effect  has  penetrated  appreciably, 
the  field  is  withdrawn  and  reestablished  in  the  opposite  sense. 


306 


MAGNETIC  TESTING 


[XII,  §  210 


Dae  to  this  effect,  iron  has  little  influence  in  increasing  the 
inductance  of  coils  in  high  frequency  circuits. 

211.  Residual  Magnetism  and  Coercive  Force.    It  was 

stated  in  §  194  that  no  energy  is  required  to  maintain  the 


B 


30 


20 


10 


OS     10     15 
FIG.  149. 


.Ztf 


SO 


magnetic  flux  when  once  it  has  been  established.  Energy  is 
required,  however,  to  increase  the  magnetic  flux,  and  energy 
is  given  out  if  the  flux  decreases. 

A  gradually   increasing   magnetizing  field   impressed  on  a 
piece  of  iron  will  cause  the  induction  density  to  rise  along 


XII,  §  212]          MAGNETIZATION   CURVES  307 

some  such  curve  as  oa,  Fig.  149.  If  H  is  now  gradually  re- 
duced in  value,  the  iron  shows  a  more  or  less  marked  tendency 
to  persist  in  its  state  of  magnetization.  For  a  given  decre- 
ment in  H,  the  decrease  in  B  is  less  than  was  the  increase  for 
the  corresponding  increment  in  H.  Hence  the  curve  returns 
along  some  such  path  as  ab,  which  is  quite  different  from 
that  along  which  it  rose  to  the  point  a.  The  intercept  ob 
represents  the  remanence,  or  the  residual  magnetism  retained 
by  the  iron. 

If  the  magnetizing  field  is  reversed  in  direction  at  this  point 
and  again  gradually  increased  in  the  negative  direction,  the 
curve  drops  along  some  such  path  as  be.  The  intercept  oc 
represents  the  coercivity  of  the  iron.  This  is  conveniently 
measured  in  terms  of  the  coercive  force,  which  is  the  value  of 
the  reversed  magnetizing  field  oc  required  to  reduce  the  resid- 
ual magnetism  to  zero. 

The  ratio  of  the  residual  magnetism  to  the  previous  maxi- 
mum value  of  B,  or  ob/oc,  is  called  the  retentivity.  Some 
writers  use  retentivity  as  synonymous  with  residual  mag- 
netism. A  closed  circuit  of  soft  iron  may  retain  85  %  of  its 
maximum  induction,  and  a  coercive  force  of  less  than  two 
gausses  is  sufficient  to  reduce  it  to  zero.  On  the  other  hand 
hardened  steel  may  require  a  coercive  force  of  40  or  50 
gausses  to  demagnetize  it. 

212.  Hysteresis.  It  has  been  shown  in  the  preceding 
article  that  changes  in  the  induction  density  always  lag  behind 
the  corresponding  changes  in  the  magnetizing  field  strength. 
This  tendency  is  called  hysteresis.  We  shall  now  show  that 
the  area  of  the  entire  loop  (Fig.  149),  is  a  measure  of  the  work 
done  in  carrying  the  sample  through  a  magnetic  cycle.  This 
energy  appears  as  heat  in  the  sample. 

In  order  to  calculate  the  work  expended  in  a  magnetic  cycle, 
assume  a  long  bar  of  iron,  or  better  a  ring,  of  length  L  centi- 


308  MAGNETIC  TESTING  [XII,  §  212 

meters  and  of  cross  section  A  square  centimeters,  overwound 
throughout  its  entire  length  with  n  turns  of  wire  per  centi- 
meter. If  the  magnetizing  current  i  is  increased  by  some 
small  amount  di  in  a  time  dt,  a  corresponding  increase  in  the 
induction  density  dB  will  be  produced.  In  accordance  with 
Lenz's  law,  this  increase  will  set  up  a  counter-electromotive 
force  of  value  E,  in  opposition  to  the  current  i.  Against  this 
counter-electromotive  force  the  current  must  do  work  whose 
value  is 
(1)  dW=Eidt. 

But  E  =  dN/dt}  where  N  is  the  total  number  of   linkings. 
Also  we  have  from  equation  (2),  §  128, 


/o\  rj  -\r . 

(Z)  CUM  - 

since  Ln  is  the  number  of  wire  turns  and  AdB  is  the  mag- 
netic flux.  If  V,  the  volume  of  the  specimen,  is  put  in  place 
of  LA,  we  may  write 

(3)  dW=  VnidB-, 

hence,  the  work  per  cubic  centimeter  is  given  by 

(4)  dW=nidB. 

Substituting  the  value  of  ni  from  equation  (33),  §  109,  (4)  may 
be  written  in  the  form 

(5)  ^=M-B. 

The  total  work  done  as  H  and  B  vary  between  assigned  limits 
is  given  by  integrating  this  expression  for  dW 

(6)  W=—  fhdB. 

Kef  erring  to  Fig.  149,  as  H  is  increased  from  zero  to  Hj,  B  is 
increased  from  zero  to  B1?  and  the  amount  of  work  expended 


XII,  §  212]          MAGNETIZATION   CURVES  309 

on  the  iron  is  given  by  the  expression 


area  aeo 


47T 

that  is  the  sum  of  all  the  small  strips  yy'.  As  H  is  brought 
back  to  zero,  B  falls  to  a  value  06,  and  the  changing  mag- 
netization returns  energy  to  the  circuit  of  value  aeb/4:  TT.  The 
net  amount  of  work  expended  on  the  iron  is  then  given  by 
area  a&o/4  TT.  Extending  this  analysis  over  the  entire  area  of 
the  loop,  we  find  that  the  work  expended  in  the  complete 
cycle  is 
/ys  -r^r  _  area  of  loop 

4  7T 

This  will  be  expressed  in  ergs  per  cubic  centimeter  per  cycle, 
if  the  area  is  taken  in  square  centimeters. 

Since  the  values  of  B  are  large  compared  with  H,  it  will 
not  be  convenient  to  plot  them  to  the  same  scale.  However, 
if  u  represents  the  number  of  H  units  corresponding  to  one 
scale  division  on  the  cross-sectioned  paper,  and  if  v  represents 
the  number  of  B  units  per  division,  then 

W=^  Cxdy 

(«)      J*:r 

=  —  area  of  loop    ergs  per  cubic  centimeter  per  cycle. 
47rL  J 

In  addition  to  the  hysteresis  loss,  there  is  also  a  loss  of 
energy  due  to  the  eddy  currents  in  the  iron  core  when  the 
iron  is  subjected  to  the  magnetizing  field  of  an  alternating 
current.  This  effect  is  greatly  diminished  by  building  the 
sample  of  thin  sheets,  insulated  from  one  another  by  varnish. 
The  discussion  of  total  core-loss  will  be  resumed  in  §  237. 

Instead  of  expressing  the  energy  loss  in  ergs  per  cubic  centi- 
meter per  cycle,  it  is  common  in  practice  to  express  the  energy 
in  watts  per  pound  or  watts  per  kilogram,  at  a  given  frequency. 
Some  typical  values  are  given  in  the  accompanying  table  for 


310 


MAGNETIC  TESTING 


[XII,  §  212 


samples  of  steel  and  iron  carried  through,  various  ranges  of 
induction  density. 


DYNAMO-MAGNET  STEKL 

TRANSFORMER  IRON 

B 

limits 

Ergs  per 
cu.  cm. 
per  cycle 

Watts  per 
Ib. 
freq.  100 

B 

limits 

Ergs  per 
cu.  cm. 
per  cycle 

Watts  per 
Ib. 
freq.  100 

2000 

550 

0.32 

2000 

240 

0.14 

5000 

2030 

1.20 

5000 

1190 

0.70 

9000 

5250 

3.09 

7000 

2020 

1.20 

12000 

8500 

5.01 

9000 

3060 

1.80 

16000 

13900 

8.20 

HYSTERESIS  Loss    (Ewing) 

The  first  hysteresis  loop  taken  on  a  neutral  sample  of  iron, 
in  which  all  effects  of  previous  history  have  been  destroyed, 
will  not  close  at  the  point  a,  Fig.  149.  Successive  loops,  how- 
ever, will  progressively  show  less  and  less  of  this  defect. 
After  repeated  reversals  (for  practical  purposes,  20),  the 
position  of  the  cusps  will  be  invariable,  and  a  strictly  cyclic 
state  will  be  established. 

The  normal  induction  curve  is  the  locus  of  the  cusps  of  a 
series  of  hysteresis  loops  which  are  strictly  cyclic.  The  shape 
of  the  hysteresis  loop  may  be  inferred  with  a  fair  degree  of 
accuracy  if  three  points  are  located :  (1)  the  extreme  cusp, 
(2)  the  residual  magnetism  intercept,  (3)  the  coercive  force 
intercept.  It  is  seen  from  the  above  table  that  the  hysteresis 
loss  varies  greatly  with  the  conditions  of  the  test.  In  order 
to  compare  results,  it  is  important  to  specify  standard  con- 
ditions. These  are  usually  taken  as  60  cycles  per  second,  and 
Bmax  ==  10000  gausses. 

Typical  hysteresis  loops  for  three  samples  of  widely  different 
materials  are  shown  in  Fig.  150  and  Fig.  151.  In  I,  Fig.  150, 
a  small  value  of  H  sets  up  a  large  induction  density  which 
may  be  easily  reversed,  the  hysteresis  loss  being  small.  Such 


XII,  §  212]          MAGNETIZATION  CURVES 


311 


material  is  useful  for  alternating  current  transformer  cores. 
A  low  hysteresis  loss  is  important  in  this  case  because  the  loss 
is  continuous,  even  during  the  time  for  which  there  is  no 
current  output  from  the  transformer.  Punching  and  shearing 


d. 


ganpses 


// 

' 


10 


FIG    150. 

of  the  material  in  the  course  of  manufacture  will  increase  the 
hysteresis  loss  somewhat,  but  this  is  in  part  overcome  by  sub- 
sequent annealing. 

In  II,  Fig.  150,  B  increases  rapidly,  the  residual  magnetism 
is   large,   and   the   material   is   easily  demagnetized.      These 


312 


MAGNETIC  TESTING 


[XII,  §  212 


properties  make  the  material  useful  for  dynamo  field  mag- 
nets. 

In  Fig.  151,  the  retentivity  and  coercive  force  are  both  high, 
which  are  desirable  qualities  for  steel  to  be  used  for  permanent 
magnets. 


B 


H 


90 


FIG.  151. 

213.  The  Hysteretic  Constant.  If  values  of  the  hysteresis 
loss  for  different  values  of  Bmuj,  expressed  in  ergs  per  cubic 
centimeter  per  cycle,  are  plotted  against  the  corresponding 
values  of  Bmaz,  a  curve  of  the  form  shown  in  Fig.  152  is  ob- 
tained. Steinmetz  has  given  an  empirical  formula  which  ap- 
proximately expresses  this  relation  ;  it  is 


XII,  §  214] 


MAGNETIZATION   CURVES 


313 


(9)  W—  rjB1-6  ergs  per  cubic  centimeter  per  cycle. 

The  factor  77  is  called  the  hysteretic  constant,  and  for  average 
samples  of  sheet  steel  its  value  may  vary  from  0.001  to  0.003. 
For  hardened  tungsten  magnet  steel  77  =  0.06,  and  for  a  high 
grade  of  silicon  steel  rj  =  0.0006. 


s-ooo- 


Bmax. 


10  kg. 


FIG.  152. 


214.  The  Temperature  Rise.  The  rise  in  temperature  of  a 
sample  of  iron  due  to  hysteresis,  when  subjected  to  a  cyclic 
magnetization,  is  calculated  as  follows.  When  all  the  heat  is 
assumed  to  be  retained  in  the  iron,  we  may  write 

W=JH 

where  H  is  the  heat  in  calories,  equivalent  to  W  ergs,  and 
where  J  is  the  mechanical  equivalent  of  one  calorie  of  heat 
expressed  in  ergs.  Consider  a  cubic  centimeter  of  the  iron  of 
specific  heat  0.115  and  of  specific  gravity  7.6.  The  heat 
accumulated  in  the  iron  is  given  by  the  product  of  the  mass, 
the  specific  heat,  and  the  rise  in  temperature ;  that  is 

H=  7.6  X  0.115  X  *. 
The  energy  equivalent  of  this  heat  is 

W=  JH=  (4.2  x  107)(7.6  x  0.115  X  *), 
which  is,  by  equation  (6),  equal  to 

-L  r 

47T*/ 


314  MAGNETIC   TESTING  [XII,  §  214 

It  follows  that  the  expression  for  the  temperature  rise  becomes 


215.  Alloys  of  Magnetic  Materials.  The  influence  of  the 
admixture  of  various  chemical  elements  on  the  magnetic 
properties  of  iron  and  steel  has  been  intensively  studied,  and 
the  adaptability  of  the  material  to  its  several  uses  has  been 
greatly  increased  by  this  means. 

The  addition  of  2.5  %  of  silicon  to  high  grade  soft  iron 
(Bmax  =  4000,  /A  =  2000)  increases  the  permeability  over  two- 
fold, while  the  hysteresis  loss  is  reduced  34  %  and  the  coercivity 
27%. 

Small  percentages  of  silicon  improve  steel  also,  but  in  a 
somewhat  less  degree,  and  at  the  same  time  render  it  less 
liable  to  deterioration  at  high  temperatures.  The  resistivity 
also  is  increased,  and  the  eddy  current  core-loss  is  reduced. 
Improved  sheet  steel  of  this  character  shows  a  core-loss  as  low 
as  0.9  watt  per  pound,  at  a  frequency  of  60  cycles  per  second, 
with  Bmaz  =  10000  gausses.  For  Bmajc  =  2000,  the  loss  falls  as 
low  as  0.05  watt  per  pound. 

The  addition  of  12  %  of  manganese  reduces  the  susceptibility 
of  steel  practically  to  zero.  Certain  alloys  containing  as  high 
as  88  °/G  of  iron  have  been  made  which  are  non-magnetic. 
Recent  experiments  have  shown  that  the  permeability  of 
electrolytic  iron  is  very  greatly  increased  when  melted  in  a 
vacuum,  while  at  the  same  time  the  hysteresis  loss  is  reduced. 
These  effects  are  accompanied  by  a  low  resistivity  which  is 
favorable  to  a  large  eddy  current  loss,  but  the  resistivity  may 
be  increased  by  the  addition  of  such  elements  as  silicon  or 
aluminum  without  materially  affecting  the  other  magnetic 
qualities. 

Further  information  on  the  influence  of  the  composition  on 


XII,  §  217]          MAGNETIZATION  CURVES  315 

magnetic  properties  will   be   found   in  the  various   electrical 
handbooks. 

216.  Alloys  of  Non-magnetic  Materials.    Many  alloys  of 
practically  non-magnetic  components  are  themselves  more  or 
less  magnetic.     An  alloy  of  25  %  manganese  and  75  %  copper, 
which  is  itself  non-magnetic,  was  rendered  strongly  magnetic 
by  decreasing  the  copper  content  and  adding  13  %  of  aluminum. 

Maganese  and  aluminum  compounds  are  in  general  magnetic, 
and  certain  of  them  approach  cast  iron  in  magnetic  quality. 
The  effect  of  the  copper  in  the  alloy  mentioned  above  appears 
to  be  solely  that  of  keeping  the  alloy  soft  enough  to  be  worked. 
Hysteresis  is  large  in  most  manganese-aluminum  alloys. 

Very  surprising  reversals  of  properties  occur  for  extremes 
of  both  heat  and  cold.  A  certain  25  °/0  nickel  steel,  practi- 
cally non-magnetic  at  ordinary  temperatures,  is  strongly  mag- 
netic at  —190°  C.,  and  retains  the  property  when  restored  to 
room  temperature.  Heating  to  600°  C.,  and,  then  cooling 
slowly,  destroys  the  magnetism  for  many  of  these  alloys. 

217.  Residual   Magnetism    and    Retentivity.     With  the 
rapid  development  of  electro-magnets,  the  permanent  magnet 
was  displaced  from  the  position  of  importance  which  it  had 
previously    occupied,  and    research   was   for  a  time  diverted 
along  lines   of   more  immediately  practical  application.      In 
recent  years,  however,  permanent  magnets  have  entered  largely 
into  the  manufacture  of  galvanometers,  quantity  meters  and 
other  measuring  instruments  of  many  kinds,  magneto-ignition 
devices,  speedometers,  toys,  and  automatic  machines  in  great 
variety.     Accordingly,  their  production  and  properties  have 
been  intensively  studied,  and  they  now  constitute  a  large  factor 
in  the  electrical  industry. 

Magnetic  materials  vary  greatly  with  regard  to  the  amount 
of  magnetism  retained  after  the  magnetizing  field  has  been 
withdrawn.  Soft  iron  has  a  high  residual  magnetism,  often 


316  MAGNETIC   TESTING  [XII,  §  217 

85  % ,  but  it  is  loosely  held,  the  coercive  force  sometimes  being 
as  low  as  two  gausses.  Hardened  steel  on  the  other  hand  has 
much  less  residual  magnetism,  but  it  is  tenaciously  held,  and 
requires  a  strong  reversed  field  of  perhaps  50  gausses  to  re- 
move it.  A  dosed  magnetic  circuit  of  soft  iron  will  show 
large  remanence  as  long  as  it  is  not  broken.  After  introduc- 
ing an  air  gap,  however,  the  iron  is  quickly  demagnetized. 

When  the  magnetizing  force  is  removed  from  a  bar  it  tends 
to  demagnetize  itself.  This  effect  is  much  greater  for  short 
bars  than  for  long  ones,  and  soft  iron  bars  for  which  the  ratio 
of  length  to  diameter  is  as  small  as  ten  show  scarcely  any 
residual  magnetism.  Bars  for  which  the  ratio  of  length 
to  diameter  is  as  great  as  400  retain  a  large  part  of  the 
induced  magnetism  after  the  magnetizing  force  has  been  with- 
drawn. 

The  total  induction  in  a  magnetic  circuit  of  iron  may  be 
regarded  as  comprising  three  components,  (a.)  the  temporary 
magnetism,  which  vanishes  with  the  removal  of  the  magnetiz- 
ing field ;  (6)  the  sub-permanent  magnetism,  which  is  removed 
by  the  polar  field  or  by  special  treatment ;  (c)  the  permanent 
magnetism,  which  can  only  be  removed  by  the  application  of 
a  sufficiently  strong  reversed  field,  perhaps  accompanied  by 
vibration. 

For  permanent  magnets  the  necessary  characteristics  are 
large  retentivity  and  coercive  force,  with  small  tendency  to 
deteriorate  with  lapse  of  time.  Steel  for  such  magnets  is 
alloyed  with  3  to  5  %  of  tungsten,  with  the  addition  of  a 
fraction  of  a  per  cent  of  chromium,  which  improves  the  sta- 
bility of  the  magnetism.  The  sub-permanent  magnetism  is 
removed  by  an  artificial  ageing  process  in  which  the  finished 
magnets  are  alternately  heated  and  cooled  in  a  water  or  oil 
bath,  and  then  subjected  to  a  rapid  mechanical  vibration  with 
the  addition  of  a  slight  demagnetizing  effect  due  to  an  alter- 
nating current  field.  This  treatment  tends  to  bring  about  the 


XII,  §  217] 


MAGNETIZATION   CURVES 


317 


same  changes  as  long  continued  use,  and  magnets  so  treated 
deteriorate  very  little  with  the  lapse  of  time. 

The  effective  strength  of  bar  magnets,  measured  in  terms 
either  of  B  or  I,  will  depend  upon  the  dimensions  of  the  bars 
and  the  shape  into  which  they  are  formed,  as  well  as  upon  the 
quality  of  the  material  used.  Tests  for  the  residual  magnetism 
of  permanent  magnets  must  be  made  on  specimens  which  are 
virtually  closed  magnetic  circuits  to  secure  results  which 
are  characteristic  of  the  material  used.  Any  test  on  a  short 
bar  will  yield  results  characteristic  of  that  length  and  shape 
of  specimen  only,  due  to  the  demagnetizing  effect  of  the  poles. 

If  a  hysteresis  loop  is  drawn  for  a  perfect  magnetic  circuit 
of  the  given  material  (Fig.  153),  the  retentivity  is  repre- 
sented by  OB!  and  the  coercive 
force  by  OHi.  The  magnetic 
circuit  then  has  an  intrinsic 
M.  M.  F.  sufficient  to  maintain 
the  induction  B15  which  is  meas- 
ured by  H!  gilberts  for  every  • 
centimeter  of  length  of  the  cir- 
cuit. An  air  gap  introduced  into 
the  circuit  will  reduce  the  re- 
manence  to  some  point  B2.  If 
LI  and  L>2,  respectively,  represent 
the  lengths  of  the  steel  and  air 

portions  of  the  circuit,  then  (H!  —  H2)^i  gilberts  is  the  mag- 
netomotive force  reserved  for  the  steel,  while  H2/^  is  the 
magnetomotive  force  required  to  maintain  the  reduced  induc- 
tion B2  across  the  air  gap.  From  such  a  curve  it  is  possible  to 
determine  in  advance  the  dimensions  of  the  magnet  for  any 
required  induction  density.  The  area  of  the  cross  section  is 
generally  chosen  so  that  the  value  of  B  is  from  2000  to  4000 
gausses,  and  the  corresponding  value  of  H  is  read  from  the 
curve,  Fig.  153. 


., 

B*  / 

/  H 


FIG.  153. 


318  MAGNETIC  TESTING  [XII,  §  217 

Corrections  in  the  form  of  empirical  constants  are  applied 
for  the  influence  of  any  joints  or  pole  pieces,  as  well  as  for 
leakage  and  polar  field.  A  large  value  of  the  ratio  (H].—  H2)/H! 
indicates  good  keeping  quality. 

218.  Testing  of  Permanent  Magnets.     It  is  seldom  that  a 
precise  measurement  of  the  actual  flux  density  in  a  bar  magnet 
is  required.     The  usual  specifications  simply  require  that  it 
shall    fall    within   certain  defined    limits,   both    of   flux   and 
of  constancy. 

A  method  for  determining  the  flux  density  is  given  in  §  236. 
If  the  magnet  is  a  straight  bar,  the  intensity  of  magnetization 
may  be  derived  from  magnetometer  readings.  If  in  the  form 
of  the  letter  U,  the  magnet  should  be  provided  with  smooth 
end  faces  across  which  a  soft  iron  armature  is  placed.  A  coil 
of  a  known  number  of  turns  is  slipped  over  the  middle  point 
and  connected  to  a  ballistic  galvanometer.  The  quick  removal 
or  replacing  of  the  armature  gives  a  throw  on  the  galvanom- 
eter which  may  be  compared  with  that  taken  in  a  similar 
way  on  a  standard  magnet.  The  total  flux  also  may  be  com- 
puted by  this  method,  using  a  calibrated  ballistic  galvanom- 
eter. 

The  given  magnet  may  also  be  compared  with  a  standard 
magnet  by  measuring  the  forces  required  respectively  to 
detach  the  armature,  or  by  comparing  the  torques  due  to  eddy 
currents,  when  a  copper  disk  is  rotated  between  the  poles. 

219.  Method  of   Current  Reversals.     In  order  to  remove 
all  traces  of   existing   magnetism  or  to   annul   any   previous 
magnetic  history,  it  is  only  necessary  to  raise  the  iron  to  a 
red   heat  and   cool  it   in   a   magnetic  field  of  zero   strength. 
However,  this  method  is  not  practicable.     A  sufficiently  effec- 
tive demagnetization  is  obtained  conveniently  by  placing  the 
test  piece  in  an  alternating  current  field  and  gradually  reduc- 
ing its  intensity  to  zero  by  introducing  series  resistance,  by 


XII,  §  219]          MAGNETIZATION   CURVES  319 

reducing  the  voltage  with  a  potentiometer  device,  or  by  shut- 
ting down  the  generator. 

Another  method  is  that  of  rapidly  reversing  a  direct  current, 
to  which  the  magnetizing  field  is  due,  and  at  the  same  time 
reducing  its  strength  by  a  gradually  increasing  series  resist- 
ance. The  current  should  be  reduced  so  that  B  decreases 
uniformly,  and  the  reversals  should  not  be  more  rapid  than 
one  per  second. 


320  MAGNETIC  TESTING  [XII,  §  220 

PART  II.     METHODS  OF  MAGNETIC  TESTING 

220.  Classification.     The  approved  methods    for  studying 
magnetic  properties  may  be  arranged  under  four  headings  as 
follows : 

(1)  Magnetometer  methods. 

(2)  Induction-ballistic  methods. 

(a)  Ring  method. 

(6)  Bar  and  yoke  methods. 

(3)  Traction  methods. 

(4)  Air  gap  methods. 

221.  The  Magnetometer  Method.     This  method  is  appli- 
cable only  to  open  or  imperfect  magnetic  circuits,  that  is,  with 
samples  having  free  poles. 

The  experimental  work  is  carried  out  with  apparatus  some- 
what like  that  used  in  §  203  for  finding  the  horizontal  com- 
ponent of  the  earth's  field. 

A  sensitive  magnetometer  is  set  up  at  a  place  where  H 
is  accurately  known,  and  the  effect  of  the  poles  of  the  mag- 
netic test  piece,  under  different  values  of  magnetizing  field, 
is  found  in  terms  of  the  deflections  of  the  magnetometer 
needle  and  H. 

The  method  is  useful  only  in  certain  lines  of  magnetic 
research  and  has  no  place  in  commercial  testing.  A  serious 
objection  to  this  method  is  that  the  shape  of  the  test  piece 
largely  influences  the  results. 

We  have  seen  that  any  bar  or  rod  exerts  a  demagnetizing 
influence  upon  itself.  This  form  of  the  test  piece  can  be  used 
without  corrections  only  when  the  length  is  300  or  400  times 
the  diameter. 

Approximate  correction  factors  can  be  computed  and  applied 
for  the  end  effects  in  the  case  of  round  or  square  bars.  With 
ellipsoidal  specimens,  absolute  and  reliable  results  can  be 


XII,  §  222]         EXPERIMENTAL   METHODS  321 

obtained,  but  such  test  pieces  are  difficult  to  prepare,  and 
offer  a  better  test  of  the  skill  of  the  mechanician  than  of 
magnetic  quality. 

Details  of  the  method  together  with  its  limitations  will  be 
found  in  the  larger  treatises  on  magnetism.  It  will  not  be 
considered  further  in  this  book. 

222.  The  Ring-Ballistic  Method.  Most  of  the  practical 
magnetic  testing  to-day  is  based  on  the  induction-ballistic 
method.  The  ring-shaped  test  piece  is  superior  because  there 
is  no  free  magnetism,  and  hence  there  are  no  poles.  We  have 
seen  that  the  presence  of  poles  exerts  a  demagnetizing  effect 
on  the  test  specimen.  The  ring  is  uniformly  overwound 
throughout  with  turns  of  a  wire  which  is  large  enough  to  carry 
the  desired  magnetizing  current  without  heating.  A  secondary 
coil  of  fine  wire  is  wound  over  the  primary,  and  is  so  arranged 
that  any  chosen  number  of  turns  may  be  connected  to  the 
ballistic  galvanometer. 

The  ring  may  be  cut  or  stamped  from  a  plate  of  the  given 
material,  or  a  bar  may  be  bent  into  a  circular  form  and  the 
ends  welded  together. 

The  radial  thickness  of  the  ring  should  be  small,  as  it  is 
found  that  the  magnetic  changes  do  not  occur  instantly,  and 
the  time  required  to  bring  about  a  change  in  the  magnetic  flux 
is  greater  as  the  thickness  increases. 

Since  the  induction  method  is  based  on  the  measurement  of 
the  quantity  of  electricity  induced  in  the  secondary  coil,  it 
follows  that  the  galvanometer  throw  may  occur  before  the 
magnetization  has  reached  its  final  value,  in  which  case  the  ob- 
served throw  will  not  represent  the  total  charge.  This  effect 
is  not  troublesome  in  thin  rings  and  disappears  at  the  higher 
values  of  flux  density.  The  liability  of  error  is  least  when 
a  ballistic  galvanometer  of  fairly  long  period  is  used. 

The  theory  of  the  method  is  as   follows.     Let   us  assume 


322 


MAGNETIC  TESTING 


[XII,  §  222 


a  circuit  arranged  as  shown  in  Fig.  154.  A  storage  battery  of 
suitable  voltage  is  connected  through  an  ammeter  A  and  a 
control  rheostat  R'  with  the  middle  points  of  a  reversing 
switch  W.  This  switch  is  connected  also  to  a  double  pole 
double  throw  switch  K.  With  the  switch  K  on  the  right  hand 
points,  current  flows  through  the  primary  or  magnetizing  coils 
of  the  ring  R.  With  the  switch  K  on  the  left  hand  points, 
current  flows  through  the  primary  coils  p  of  a  known  mutual 
inductance  M.  The  secondary  coils  of  R  and  M  are  connected 
in  series  with  a  ballistic  galvanometer  g,  and  a  resistance  box 

r'.    The  total  resist- 
M  S~\       K   S^ xX  ance    °f    the    com- 

'  n    '     -"""^      '  s  x  N 

bined  secondary 
circuits  will  be 
kept  constant 
when  once  a  d- 
justed.  The  pri- 
mary coils  on  the 
ring  must  be  closely 
and  uniformly 
wound  over  the 
entire  length  of 
the  iron  core.  The 
secondary  coils 
may  be  bunched 
at  any  convenient 
place  on  the  ring. 

With  K  thrown  to  the  ring  side,  a  reversal  of  W  will  reverse 
a  current  /  in  the  ring  primary.  This  withdraws  and  reestab- 
lishes the  flux  through  the  secondary  coil,  which,  by  linking  with 
$  wire  turns,  induces  a  charge  Qi  in  the  galvanometer  circuit. 
Let  AjVi  represent  the  corresponding  total  change  in  the 
linkings  of  flux  and  wire  turns.  Then  we  have,  by  equations 
(76),  §  142  and  (18),  §  149, 


XII,  §  222]         EXPERIMENTAL  METHODS  323 

(11)  Ql 

where  D  is  the  galvanometer  throw  which  occurs  when  the 
switch  is  reversed,  r  is  the  total  resistance  of  the  secondary 
circuit,  and  G  is  the  ballistic  constant  of  the  galvanometer. 
By  equation  (2),  §  128,  we  have  also 

(12)  JVi  =  *S, 
which  may  be  written  in  the  form 

(13)  N^BAS, 

where  B  is  the  induction  density  iL  the  iron  and  A  is  the  area 
of  cross  section  of  the  ring.  Any  change  in  JVi  is  due  to  a 
change  in  B  ;  hence  the  expression  AJVi/r  becomes 


and  equation  (11)  may  be  written  in  the  form 
(14) 


Since  there  are  two  unknown  quantities,  AB  and  G,  in  this 
equation,  it  is  necessary  to  have  another  equation  which  gives 
the  value  of  G,  in  order  to  calculate  AB.  If  K  is  now  thrown 
to  the  mutual  inductance  side,  a  reversal  of  W  will  reverse  the 
current  of  strength  i  through  the  primary  of  the  mutual  in- 
ductance of  value  M,  and  the  induced  charge  in  the  secondary 
will  cause  a  deflection  d  on  the  galvanometer.  The  charge  Q2 
induced  by  this  reversal  of  the  current,  is  given  by  equation 
(18),  §  149,  and  is 

(15)  Q2  =  2Mi=Gd. 


324  MAGNETIC   TESTING  [XII,  §  222 

Eliminating  G  between  this  equation  and  (14),  and  solving  for 
AB,  we  find 

(16)  AB=2^. 

ASd 

Since  A,  S,  and  M  are  constants,  and  since  i/d  is  a  constant 
for  the  mutual  inductance  used,  it  is  convenient  to  write 

(17)  AB  =  KD. 

It  is  frequently  found  that  the  deflection  D  is  either  too 
small  or  too  large  for  convenience,  or  it  is  not  of  the  same 
order  of  magnitude  as  d.  Values  of  D  and  d  can  be  controlled 
by  changing  r'.  However,  if  the  total  secondary  resistance  is 
not  kept  constant  throughout,  it  becomes  necessary  to  measure 
the  secondary  resistances  for  each  case,  and  the  calculations 
are  somewhat  longer. 

A  more  convenient  way  to  control  the  value  of  D  is  to  have 
a  variable  number  of  turns  in  the  ring  secondary,  each  coil, 
however,  having  the  same  resistance,  so  that  the  total  resist- 
ance remains  constant  as  the  number  of  turns  is  changed.  In 
case  it  is  found  necessary  to  change  the  number  of  turns 
during  the  progress  of  an  experiment,  equation  (16)  may  be 
written  in  the  form 


When  current  is  flowing  through  the  primary  of  the  ring, 
the  reversal  of  this  current  causes  the  collapse  of  the  mag- 
netizing field  H,  and  the  establishment  of  a  numerically  equal 
field  in  the  opposite  direction.  This  change  in  H  causes  a 
corresponding  reversal  in  B.  By  reference  to  Fig.  149  it 
will  be  seen  that  if  H  is  changed  from  some  positive  value  at 
a  to  the  corresponding  negative  value  at  a',  the  change  in  B  is 
equal  to  twice  the  value  of  B  corresponding  to  the  value  of  H 
calculated  from  the  current  read  on  the  ammeter.  This  gives 

AB  =  2  B  ; 


XII,  §  222]          EXPERIMENTAL  METHODS  325 

hence,  putting  this  value  in  equation  (18),  we  find 


This  equation  gives  B  in  C.  Gr.  S.  units,  or  gausses,  provided  M 
and  i  are  in  C.  Gr.  S.  units.  If  M  is  in  henrys  and  i  is  in 
amperes,  we  shall  have 


(20)  B  =  108          gausses 


The   value   of   the   magnetizing   field    H    for  a   current  of 
strength  /  amperes  is  given  by  the  equation 

H=ri  IT  I]/ gausses, 


where  T  is  the  total  number  of  turns  in  the  primary,  and  L  is 
the  mean  circumference  of  the  ring. 

If  the  magnetizing  current  on  the  ring  is  brought  to  any 
desired  maximum  value  /  and  is  then  reduced  to  zero,  the 
galvanometer  throw  is  not  a  measure  of  the  maximum  value  of 
B,  but  is  proportional  to  the  difference  between  the  maximum 
value  of  B  and  the  residual  induction  remaining  after  /  is  zero. 
(See  5,  §  223.)  It  must  be  remembered  that  in  this  and  the 
following  experiments,  the  galvanometer  throw  is  a  measure  of 
the  change  in  the  induction.  The  above  method  of  reversals  is 
used  in  order  to  avoid  the  influence  of  residual  magnetism. 

For  a  neutral  test  piece,  different  values  of  B  will  result 
by  suddenly  or  slowly  increasing  H.  Moreover,  the  first  few 
successive  reversals  of  H  will  not  yield  the  same  value  of  B. 
After  several  reversals,  say  twenty,  B  becomes  constant,  and 
then  its  value,  computed  from  half  the  throw,  gives  the  normal 
induction  density.  The  locus  of  several  such  points,  for  a 
suitable  range  of  H  values,  is  called  the  normal  induction  curve 
(§  212).  This  is  the  curve  that  always  should  be  used  in 
specifying  magnetic  quality. 


326  MAGNETIC   TESTING  [XII,  §  223 

223.  Laboratory  Exercise  L.  To  determine  the  magnetic 
quality  of  a  sample  of  iron  by  the  ring-ballistic  method,  with 
current  reversals. 

APPARATUS.  Iron  or  steel  test  ring  with  two  windings, 
ballistic  galvanometer,  standard  mutual  inductance,  double 
pole  double  throw  switch,  reversing  switch,  tap  key,  ammeter, 
adjustable  resistance,  control  rheostat,  and  a  few  cells  of 
storage  battery. 

PROCEDURE.  (1)  From  the  primary  ring  constants,  compute 
the  value  of  I  for  the  desired  maximum  value  of  H.  With 
the  circuit  arranged  as  in  Fig.  154,  throw  K  to  the  ring  side, 
set  r'  at  some  high  trial  value,  say  10,000  ohms,  and  note  the 
deflection  when  W  is  reversed,  using  10  secondary  turns. 
Reduce  r'  until  the  reversal  of  /  gives  a  full  scale  deflection. 
Keep  this  value  of  r'  unchanged  throughout  the  experiment. 

(2)  Carefully  demagnetize  the  ring.     In  doing  this,  make  7 
slightly  greater  then  in  (1),  and  rock  the  reversing  switch  with 
the  galvanometer  circuit  open,  meanwhile  bringing  the  current 
to  zero  by 'means  of  R'.     (See  §  219.) 

(3)  Throw  K  to  the  mutual  inductance  side  with  k  open, 
and  note  the  zero  position  of  the  galvanometer.     Close  Jc  and 
set  the  rheostat  R1  so  that  the  current  i,  when  reversed  through 
the   primary  of   3f,   gives  a  full    scale   deflection.      Eeverse 
several  times,  and  read  and  record  values  of  i  and  d.     Eepeat 
these  readings  for  values  of  i  approximately  one  half  and  one 
third  as  large  as  before,  and  calculate  the  ratio  of  the  current 
to  the  mean  value  of  the  corresponding  deflections.     This  gives 
the  value  of  i/d  in  equation  (20). 

(4)  Throw  K  to  the  ring  side  with  W  open.     Close  W  with 
R'  set  at  its  maximum  value.     Then  begin  reducing  R1  until 
H  is  only  a  few  units,  say  two  or  three  gausses,  and  reverse  the 
current  about  20  times,  having  previously  opened  the  galva- 
nometer circuit.     Close  the  galvanometer  circuit  and  reverse 
the  current  again,  this  time  reading  the  throw  D.     Diminish 


XII,  §  223]         EXPERIMENTAL  METHODS  327 

R'j  bringing  the  current  to  a  slightly  higher  value,  reverse 
about  20  times  as  before,  and  then  read  the  throw  for  the  next 
reversal.  These  reversals  are  necessary  to  establish  a  strictly 
cyclic  state  in  the  magnetism  of  the  ring.  During  these 
reversals  the  galvanometer  circuit  must  be  open.  A  tap  key 
may  be  used  for  this  purpose. 

Continue  the  above  procedure  for  12  or  15  steps,  until  the 
desired  maximum  value  of  H  is  reached.  The  instructor  will 
advise  concerning  the  maximum  values  of  H  and  B  for  the  test 
piece  used.  The  first  few  steps  should  be  small  ones,  because 
here  the  B-H  curve  changes  its  slope  most  rapidly  for  small 
values  of  B  and  H.  Readings  thus  taken  after  several  re- 
versals may  be  repeated  as  a  cheek.  In  case  it  is  necessary  to 
repeat  a  reading  for  a  value  of  H  which  is  less  than  the  one 
before,  the  test  piece  must  be  demagnetized  again. 

(5)  In  order  to  study  the  residual  magnetism  in  the  ring, 
proceed  as  follows.  After  taking  the  reading  of  the  throw  for 
each  reversal  of  the  current,  bring  the  reversing  switch  to  its 
middle  position,  thus  breaking  the  circuit.  A  throw  Dr  will 
then  be  observed  which  is  proportional  to  the  difference 
between  the  value  of  B  and  the  value  of  the  residual  induction 
density  Br.  This  will  be  clear  from  a  study  of  Fig.  149. 

If  the  magnetic  condition  of  the  iron  at  any  instant  is  rep- 
resented by  the  point  B1?  reducing  I  (and  hence  H)  to  zero  will 
bring  the  induction  density  back  to  some  point  Br.  The  ac- 
companying galvanometer  deflection  Dr  is  then  proportional  to 
B,  —  Br.  In  order  to  compute  the  value  of  Br  which  cor- 
responds to  the  original  magnetizing  field  Hj  it  is  necessary  to 
subtract  Dr  from  the  throw  corresponding  to  Bly  that  is,  from 
half  the  throw  for  a  complete  reversal  of  I  or  D/2.  Substi- 
tuting D/2  —  Dr  in  equation  (20),  we  find  the  value  of  the 
residual  induction  density  Br  which  corresponds  to  the  pre- 
vious excitation  H!.  Immediately  after  throwing  over  the 
switch  W  and  reading  D  as  described  in  (4)  above,  quickly 


328 


MAGNETIC  TESTING 


[XII,  §  223 


open    the    double   pole   double   throw   switch,  and   read   the 
throw  Dr. 

(6)  After  the  set  of  readings  on  the  ring  is  complete  it  is 
advisable  to  take  another  set  of  calibration  readings,  as  de- 
scribed under  (3)  above.     The  mean  of  the  two  sets  of  calibra- 
tion results  should  be  used. 

(7)  The  constants  of   the  ring  and   the  mutual  inductance 
will  be  found   marked   on   the   apparatus.      The   other   data 
should  be  arranged  somewhat  as  follows. 

CALIBRATION  DATA 


i 

THROW 

MEAN  THROW 

i/d 

RING  DATA 


(8)  Calculate  first  the  calibration  ratio  i/d,  and  compute 
once  for  all  the  values  of  the  constants  in  equations  (20)  and 
(21).  Using  the  values  of  /,  calculate  the  corresponding 
values  of  H.  With  the  values  of  D  and  Dr,  calculate  the 
values  of  B  and  Br. 

It  must  be  remembered  that  half  of  the  throw  D  is  the 
measure  of  the  induction  density  B.  The  factor  2  may  be 
avoided  by  reversing  the  current  also  during  the  calibration. 
Then  d  is  twice  as  great  as  for  a  single  make  or  break.  Enter 
all  these  values  in  the  table  and  calculate  the  ratio  //,  =  B/H. 


XII,  §  224]         EXPERIMENTAL  METHODS  329 

(9)  Choose  scales  appropriate  for  the  cross-section  paper  at 
hand  and  plot  the  B-H  curve,  using  values  of  H  in  gausses  as 
abscissas.     Values  of  B,  expressed  either  in  gausses  or   kilo- 
gausses,  should  be  plotted  as  ordinates.     On  the  same  sheet,  and 
to  the  same  scale,  plot  the  Br-H  curve.     On  a  separate  sheet 
plot  the  /A-B  and  the  /x-H  curves.     A  B-H  curve  should  also 
be  plotted  with  H  expressed   in  ampere-turns   per  inch,  and 
B  in  lines  per  square  inch. 

(10)  As  the  value  of  the  current  increases,  the  galvanometer 
throws  increase,  and  may  exceed  the  limits  of  the  scale.     The 
ring  is  provided  with  a  variable  number  of  secondary  turns, 
and  the  throw  can  be  controlled  by  using  fewer  turns.     Chang- 
ing the  number  of  turns  does  no£  change  the  total  secondary 
resistance,  because  compensating  series  coils  are  introduced. 

(11)  For  low  values  of  the  magnetizing  force  the  previous 
magnetic  treatment  of  a  specimen  has  considerable  influence  on 
the  values  of  B.     This  is  the  reason  why  complete  demagnet- 
ization is  necessary.     For  higher  values  of  H,  the  values  of  B 
taken  after  several  reversals  seem  to  be  the  same,  whether  or 
not  the  ring  was  demagnetized. 

224.  The  Fluxmeter.  Instead  of  a  ballistic  galvanometer, 
an  instrument  called  the  flux  meter  is  frequently  used  for 
determining  either  induction  density  or  total  flux.  This  is 
essentially  a  suspended  coil  galvanometer,  characterized  by 
a  strong  and  uniform  magnetic  field,  negligible  torsion  in  the 
suspension  fiber,  and  excessive  electromagnetic  damping. 
The  motion  of  the  indicator  is  not  impulsive,  but  follows  the 
changing  flux,  the  limit  of  its  motion  being  a  measure  of  the 
total  charge  which  passes  through  its  coil.  The  deflection  of 
the  suspended  system  is  independent  of  the  rate  of  change 
of  the  charge.  When  used  with  a  given  test  coil,  its  scale  may 
be  graduated  to  read  directly  either  in  units  of  total  flux,  or 
of  flux  density.  A  precise  zero  setting  is  not  important,  since 


330 


MAGNETIC   TESTING 


[XII,  §  224 


readings  can  be  taken  by  observing  differences  in  the  posi- 
tion of  the  indicator.  The  fluxmeter  scale  is  calibrated  by 
means  of  a  given  test  coil  and  a  magnetizing  field  of  known 
strength. 

It  has  been  pointed  out  previously  that  changes  in  the  mag- 
netic flux  do  not  occur  instantaneously  in  iron,  especially  in 
thick  samples,  but  require  a  certain  time  for  their  completion. 
If  this  time  is  comparable  to  the  time  of  throw  of  a  ballistic 
galvanometer,  the  throw  does  not  measure  the  full  change  in 
the  flux.  The  fluxmeter,  however,  accurately  follows  the 
changes  in  the  flux,  and  it  is  under  such  conditions  that  the 
instrument  is  chiefly  used.  It  also  has  the  advantage  of  being 
direct  reading. 

225.  Bar  and  Yoke  Methods.  The  labor  and  time  required 
to  prepare  and  wind  rings  of  magnetic  material  have  led 
experimenters  to  give  much  thought  to  methods  which  would 
permit  the  use  of  short  cylindrical  bars,  which  are  easily  pre- 
pared and  for  which  the  magnetizing  and  test  coils  may  be 
wound  once  for  all. 

An  early  method  is  illustrated  in  Fig.  155.  A  massive  yoke 
YY  of  soft  iron  is  forged,  and  holes  are  drilled  at  AA', 

through  which  the 
rod  R  slips  with  the 
least  possible  clear- 
ance. Primary  and 
secondary  coils  are 
wound  on  spools 
which  slip  over  the 
portion  of  the  bar 
within  the  yoke, 
and  the  test  is  carried  through  as  in  the  case  of  the  ring. 
If  the  yoke  is  made  from  highly  permeable  material,  the 
error  introduced  by  neglecting  its  reluctance  is  not  great,  and 


FIG.  155. 


XII,  §  225]         EXPERIMENTAL   METHODS  331 

the  length  of  the  test  bar  between  the  inner  yoke  faces  may  be 
taken  as  the  length  of  the  magnetic  circuit.  However,  no 
matter  how  closely  the  bar  may  fit  in  the  holes,  there  is  a 
slight  air  gap  at  the  joints  which  introduces  relatively  great 
reluctance.  These  errors  will  prove  least  troublesome  when 
the  bar  itself  has  a  high  reluctance.  For  very  permeable 
material  they  are  too  large  to  be  neglected.  The  effect  of  the 
reluctance  of  the  yokes  and  joints  is  to  shear  the  B-H  curve 
away  from  the  B  axis,  and  approximate  corrections  may  be 
determined. 

In  a  further  modification  of  the  method,  the  bar  is  carefully 
faced  on  the  end  A'  and  made  to  abut  against  a  faced  surface 
inside  of  the  yoke.  When  magnetized,  a  force  will  be  re- 
quired to  separate  the  bar  from  the  yoke,  and  this  force  is  a 
measure  of  the  induction  density  in  the  bar.  From  the  pull 
at  C  necessary  to  detach  the  rod  from  the  yoke  the  value  of 
B  may  be  calculated. 

In  the  original  form  of  the  apparatus  as  used  by  Hopkin- 
son,  the  procedure  was  as  follows.  The  test  bar  was  divided 
at  its  middle,  and  surrounding  this  plane  of  division  was 
placed  a  test  coil  of  a  suitable  number  of  turns,  its  terminals 
being  connected  to  a  ballistic  galvanometer.  The  spool  carry- 
ing the  magnetizing  windings  was  divided,  with  a  proper 
space  between  its  inner  ends  to  permit  the  test  coil  to  pass 
through.  On  pulling  out  the  left-hand  half  of  the  rod  by  a 
force  applied  at  C,  the  test  coil  was  thrown  out  to  one  side  by 
a  spring.  The  magnetic  flux  between  the  two  portions  of  the 
bar  was  then  cut  by  the  test  coil,  giving  a  throw  on  the  ballistic 
galvanometer  from  which  B  was  calculated. 

None  of  the  methods  mentioned  above  are  capable  of  giving 
other  than  approximate  results,  and  it  remained  for  Ewing  to 
describe  a  modification  which  represented  a  great  advance 
from  the  standpoint  of  convenience  of  manipulation,  as  well 
as  accuracy  of  results. 


332 


MAGNETIC  TESTING 


[XII,  §  226 


226.  Ewing's  Double  Bar  and  Yoke  Method.  This  method 
requires  two  test  bars  of  the  given  material  and  two  yokes  of 
soft  iron.  Magnetizing  and  test  coils  are  wound  on  brass 
spools  which  can  be  slipped  over  the  bars.  A  cross  section  of 
the  arrangement  is  shown  in  Fig.  156.  The  yokes  YY'  are 
drilled  to  fit  the  standard  sized  bars,  which  may  be  firmly 
clamped  in  place  by  set  screws  in  each  yoke.  A  wooden  box 
contains  the  spools  with  the  magnetizing  and  test  coils.  A 
small  switch  playing  over  brass  studs  enables  a  variable  num- 


B( 


ooboooooooooooo 

Y 

y 

OOOO  0  QOOOO  OOOOO 
\   ocooooo 

ooooooooooooooo 

coooooooooooooo 

<            T            •* 

FIG.  15(5. 

ber  of  turns  to  be  included  in  the  test  coil,  while  the  resist- 
ance is  kept  constant. 

The  condition  of  a  perfect  magnetic  circuit  is  practically 
attained,  and  if  the  apparatus  is  used  in  the  same  manner 
as  the  ring  in  Experiment  L,  §  223,  data  for  a  B-H  curve 
may  be  taken  and  plotted,  as  shown  in  curve  1,  Fig.  157. 
This  curve  is  not  the  true  B-H  curve  for  the  material  of  the 
bars  alone,  as  no  correction  has  been  made  for  the  reluctance 
of  the  yokes  and  joints.  If  the  free  length  of  the  bars  is 
reduced  one  half,  however,  by  substituting  for  the  magnetizing 
coils  others  half  as  long  and  pushing  the  yokes  together, 
another  test  carried  through  in  the  same  manner  as  before  will 
yield  a  B-H  curve  which  falls  slightly  below  the  former,  as 
shown  in  2,  Fig.  157.  The  true  curve  for  the  bars  alone, 
corrected  for  yoke  and  joint  reluctance,  can  be  found  in  the 
following  way. '  For  any  chosen  value  of  B,  set  back  the  value 


XII,  §  226]         EXPERIMENTAL  METHODS 


333 


of  H  for  curve  1  by  an  amount  equal  to  the  distance  between 
curves  1  and  2.  Eepeat  this  process  for  a  sufficient  number 
of  points,  and  the  smooth  curve  drawn  through  these  points, 
curve  3,  Fig.  157,  will  be  the  true  B-H  curve  for  the  bars. 


3  1 


FIG.  157. 

In  the  proof  below,  we  shall  use  the  following  notation : 

H!  is  the  magnetizing  force  in  the  entire  circuit  for  the  full 
length  of  bar. 

H2  is  the  magnetizing  force  in  the  entire  circuit  for  half 
length  of  bar. 

H  is  the  magnetizing  force  in  the  bars  alone. 

LI  is  twice  the  free  distance  between  the  yokes,  1st  position. 

Lz  is  twice  the  free  distance  between  the  yokes,  2d  position. 

L,  =  2  L2. 

NI  is  the  total  number  of  magnetizing  turns  on  L^  1st  posi- 
tion. 

N2  is  the  total  number  of  magnetizing  turns  on  L2,  2d  posi- 
tion. 

/!  is  the  magnetizing  current  when  LI  is  used. 

72  is  the  magnetizing  current  when  L2  is  used. 

If  H  is  the  true  magnetizing  force  in  the  material  of  the 
bars  alone,  then  HZq  and  H7v2,  respectively,  will  be  the  true 
magnetomotive  forces  in  the  bars  in  the  two  cases.  The  total 
magnetomotive  force  around  the  entire  circuit  is  4  TT  times  the 


334  MAGNETIC  TESTING  [XII,  §  226 

number  of  ampere  turns.  Some  part  of  this,  which  may  be 
represented  by  M,  is  required  to  maintain  the  magnetic  flux 
through  the  yokes  and  joints.  It  is  assumed  that  M  remains 
constant  throughout.  The  expressions  for  the  magnetomotive 
forces  for  full  and  half  length  of  the  bars,  respectively,  may 
be  written  in  the  form 


(22) 

(23)  ^irtf/^HZa, 

Each  of  these  magnetomotive  forces  is  equivalent  to  the  sum 
of  two  components  :  one  part  is  required  to  maintain  the  flux 
through  the  bars  alone,  and  one  part  is  required  to  maintain 
the  same  flux  through  the  yokes  and  joints.  Then  we  may 
write 

(24)  HjL!  =  HA  +  M, 

(25)  H2L2  =  HZ,2  +  M. 
Subtracting  (25)  from  (24),  we  have 

(26)  H!/,!  -  H2L2  =  H[A  -  /«]. 
Introducing  the  condition  that  L±  =  2  L2,  we  find 

(27)  2  H^  -  H2L2  =  H£2, 
whence 

(28)  2H1-H2=H. 
This  may  be  written  in  the  form 

H  =  H1-H2+H1, 
or 

(29)  H  =  H1-[H2-HJ. 

The  true  value  for  H  in  the  bars  for  a  chosen  value  of  B  is, 
therefore,  given  by  subtracting  from  H]  the  difference  between 
Ho  and  H^  as  shown  in  curve  3,  Fig.  157. 


XII,  §  228]         EXPERIMENTAL  METHODS  335 

227.  Laboratory  Exercise  LI.  To  determine  the  B  —  H  curve 
of  a  sample  with  the  Ewing  double  bar  and  yoke  method. 

APPARATUS.  The  same  as  for  the  ring  method  of  Labora- 
tory Exercise  L,  §  223,  with  the  ring  replaced  by  the  double 
bar  and  yoke  equipment. 

PROCEDURE.  (1)  Arrange  the  circuit  exactly  as  for  the  ring 
method,  with  the  magnetizing  coils  in  the  battery  circuit  and 
with  the  test  coils  in  the  galvanometer  circuit.  The  bars  must 
be  carefully  demagnetized  before  beginning  each  test. 

Clamp  the  bars  in  place,  first  using  the  long  magnetizing 
coils.  Do  not  force  the  clamp  screws,  but  set  them  up  snugly. 
If  the  rods  do  not  pass  freely  through  the  holes  in  the  yokes, 
rub  them  clean  and  add  a  trace  of  vaseline  to  the  cleaning  cloth. 

Never  pound  the  rods  on  the  ends,  as  this  will  certainly  ruin 
the  fit. 

(2)  With  the  number  of  turns  in  the  secondary  windings 
suitably  chosen,  proceed  with  the  observations  precisely  as  in 
the  ring  method,  first  calibrating  the  galvanometer,  then  taking 
the  first  readings  on  the  iron  with  small  value  of  H,  not  exceed- 
ing two  or  three  gausses.     Carry  the  magnetizing  current  up  to 
the  desired  maximum,  rocking  the  reversing  switch  several  times 
with  the  galvanometer  disconnected,  before  taking  each  reading. 

(3)  Demagnetize  the  bars  again,  replace  the  long  magnetiz- 
ing coils  by  the  short  ones,  and  repeat  the  set  of  readings  with 
the  free  bar  length  reduced  one  half. 

(4)  Calculate  from  each  set  of  data  the  values  of  B  and  H 
from  the  formulas  of  §  222,  and  plot  the  curves  1  and  2,  Fig. 
157,  on  the  same  sheet.     Set  back  the  points  of  1  according  to 
the  directions  given  in  §  226,  and  draw  curve  3.     This  will  be 
the  true  B  —  H  curve  for  the  material  of  which  the  bars  are 
made.     The  precision  of  the  method  is  high  if  care  is  taken  in 
making  the  observations. 

228.   The  Compensated  Permeameter.     It  has  been  shown 
in  the  foregoing  articles  that  a  short  cylindrical  bar  is  not  a 


336  MAGNETIC  TESTING  [XII,  §  228 

satisfactory  test  piece  because  of  the  polar  field  developed  by 
its  own  magnetic  state.  However,  such  a  bar  is  a  convenient 
form  of  test  piece  because  of  the  ease  of  manufacture  and  the 
economy  of  material,  and  it  is  desirable  to  use  it  in  some  form 
of  bar  and  yoke  device.  In  the  Ewiiig  apparatus,  §  226,  the 
magnetomotive  force  required  to  overcome  the  reluctance  of 
the  yokes  and  joints  was  eliminated  by  a  double  set  of  observa- 
tions and  curves.  The  primary  objection  urged  against  this 
method  is  that  the  correction  factor  may  not  be  constant  for 
both  settings  of  the  yokes. 

In  the  compensated  permeameter,  as  described  by  Burrows, 
a  magnetomotive  force  is  supplied  by  means  of  auxiliary  cur- 
rent coils,  which  is  just  sufficient  to  maintain  the  existing  flux 
through  the  yokes  and  joints.  This  insures  a  uniform  flux 
through  the  entire  circuit,  and  practically  amounts  to  short- 
circuiting  the  bar  by  a  path  of  zero  reluctance.  The  general 


5 

Y2 

•TF 

C 

E 

F'L 
C 

FIG.  158. 

arrangement  of  the  bars,  yokes,  and  coils  is  shown  in  Fig.  158. 
The  test  bar  T,  and  an  auxiliary  bar  T2,  of  similar  but  not 
necessarily  the  same  material,  are  clamped  in  the  yokes  Y1 Y2. 
Both  bars  are  inclosed  by  uniformly  wound  magnetizing  coils, 
which  fill  the  entire  distance  between  the  yokes.  These  coils 
are  not  shown  in  the  figure. 

The  four  compensating  coils  C  are  placed  near  the  ends  of 
the  bars,  and  are  connected  in  series.  A  current  is  sent  through 
them  of  such  a  value  that  a  magnetomotive  force  is  supplied 
which  is  sufficient  to  overcome  the  reluctance  of  the  joints  and 
yokes.  Over  the  middle  of  each  bar  at  D  and  E  there  are 
wound  secondary  or  test  coils  of  100  turns  each.  At  F  and  F' 


XII,  §  228]         EXPERIMENTAL  METHODS  337 

on  bar  2\  are  two  coils  of  50  turns  each  which  may  be  con- 
nected in  helping  series  and  opposed  to  either  D  or  Et  or  they 
may  be  connected  in  opposing  series  so  that  the  charge  in- 
duced in  one  of  them  may  annul  that  in  the  other.  It  has 
been  shown  that  when  the  effect  of  coils'  F  and  F1  in  helping 
series  is  exactly  the  same  as  that  of  coil  D  or  E,  the  magnetic 
flux  may  be  taken  as  uniform  around  the  entire  circuit,  and  the 
test  bar  may  be  treated  as  though  devoid  of  poles. 

The  procedure  for  determining  a  point  on  the  B-H  curve 
for  a  given  sample  is  as  follows.  After  calibrating  the  bal- 
listic galvanometer  in  the  usual  way  with  a  known  mutual 
inductance,  the  iron  circuit  is  demagnetized.  A  small  current 
is  then  passed  through  the  magnetizing  coil  on  bar  T^ ;  from 
this  the  first  value  of  H  may  be  computed.  The  secondary 
coils  D  and  E  are  connected  in  series  with  the  ballistic  galva- 
nometer, and  are  opposed  inductively.  Current  is  now  passed 
through  the  magnetizing  coil  on  the  bar  T2  and  increased  in 
value  until,  when  both  currents  are  reversed  simultaneously, 
no  galvanometer  throw  occurs.  This  shows  that  the  magnetic 
flux  through  the  two  coils  is  the  same. 

Current  is  now  passed  through  the  four  compensating  coils 
in  series,  and  the  two  coils  F  and  F',  are  connected  in  helping 
series,  but  in  opposition  to  D.  The  three  magnetizing  currents 
are  now  reversed  simultaneously  by  means  of  a  special  gang 
switch,  and  the  compensating  current  is  adjusted  until  no 
galvanometer  deflection  can  be  observed.  The  test  bar  is  now 
in  a  uniform  state  of  magnetization,  and  the  B  value  may  be 
computed  as  in  Laboratory  Exercise  L,  §  223.  By  increas- 
ing the  magnetizing  current  on  the  bar  7\  and  repeating  the 
procedure  as  above  outlined,  other  points  on  the  B-H  curve 
may  be  obtained.1 

1  An  extended  treatment  of  this  method  is  given  in  the  following  publica- 
tions of  U.  S.  BUREAU  OF  STANDARDS:  Bulletin,  Vol.  6,  p.  31  ;  Circular  No. 
17,  Magnetic  Testing. 


338 


MAGNETIC  TESTING 


[XII,  §  229 


229.   Traction  Methods.     If  a  magnetic  circuit  includes  a 
narrow  air  gap,  free  poles  will  exist  at  the  faces  of  this  gap, 
and  there  will  be  an  attraction  between  these 
faces  which  is  given  by  the  equation 


N 


(30) 


B*A 


FIG.  159. 


where  A  is  the  face  area  and  B  is  the  induction 
density  in  the  gap.  The  force  will  be  in  dynes 
when  B  and  A  are  in  C.  G-.  S.  units. 

This  relation  can  be  proved  in  the  following 
way.  Let  N  and  S  (Fig.  159)  represent  the  two  pole  faces  just 
after  separation.  If  there  is  a  flux  density  B  uniformly  dis- 
tributed over  the  pole  faces,  we  have,  by  equation  (9),  §  192, 

(31)  BA  =  4  TTW, 

whence  the  equivalent  pole  strength  of  either  face  is  given  by 

BA 


(32) 


m  = 


The  total  flux  across  a  very  narrow  air  gap  separating  the 
attracting  poles  may  be  considered  as  made  up  of  a  component 
flux  coming  from  and  belonging  to  one  of  the  poles,  and  an- 
other component  flux  in  the  same  direction  belonging  to  and 
going  into  the  other  pole.  Hence  we  may  consider  one  pole 
of  strength  m  units,  in  a  field  of  B/2  units  due  to  the  other 
pole.  Hence  the  force  that  acts  is  given  by  the  equation 


(33) 


raB 

2 


Substituting  the  value  of  m  from  (32)  in  (33),  we  have 


(34) 


P= 


XII,  §  229]         EXPERIMENTAL  METHODS 


339 


This   gives   the   pull   in  dynes   at   the  instant  of  separation. 
Solving  (34)  for  B,  we  obtain  the  formula 


(35) 


Various  designs  of  apparatus  for  using  the  above  relation 
have  been  suggested,1  the  one  described  below  being  due  to 
Fischer-Hinnen.  Referring  to  Fig.  160,  the  principle  of  the 
method  will  be  apparent.  The  long  bar  E  rocks  freely  about 
a  point  0.  The  bar  is  graduated  in  centimeters  and  milli- 
meters, and  carries  a  sliding  weight  W  of  known  mass.  A 
counterweight  W  is  adjusted  to  bring  the  bar  into  equilibrium 
when  W  is  at  its  zero  position.  The  pull  P  takes  place  at  the 
contact  face  c,  when  current  is  passed  through  a  magnetizing 
coil  which  surrounds  the  test  bar  T.  The  weight  TPis  moved 


FIG.  160 

out  along  the  graduated  bar  E  through  a  distance  D  until  the 
separation  of  the  faced  surfaces  at  c  is  effected.  Equating  the 
moments  of  the  forces  at  this  instant,  we  find 


(36) 

whence 
(37) 


WD  =  Ph, 


1  See  STANDARD  HANDBOOK    FOR    ELECTRICAL    ENGINEERS   (4th  ed.), 
page  188. 


340  MAGNETIC  TESTING  [XII,  §  230 

Equation  (35)  may  then  be  written  in  the  form 

(38)  B 

In  order  to  avoid  facing  the  end  of  the  test  bar  it  is  fre- 
quently provided  with  a  cap  that  has  a  faced  surface.  If  the 
area  of  the  contact  face  of  the  cap  is  represented  by  A,  and 
the  area  of  cross  section  of  the  test  bar  by  As,  the  expression 
for  the  total  flux  becomes 

(39)  <£  =  BA  =  BXAS. 

The  induction  density  for  the  test  bar  is  then  given  by  the 
formula 


230.  Laboratory  Exercise  LII.  To  determine  the  B-H  curve 
of  a  sample  of  iron  with  the  Fischer-Hinnen  traction  permeameter. 

APPARATUS.  Permeameter  with  accessories,  ammeter,  re- 
versing switch,  rheostat,  and  a  few  storage  cells. 

PROCEDURE.  (1)  Arrange  the  circuit  as  shown  in  Fig.  161, 
where  C  is  the  magnetizing  coil  about  the  test  piece.  The 
rocking  bar  should  be  horizontal,  and  the  face  of  the  soft  iron 


Q 


FIG.  161. 

cap  and  the  under  face  of  the  bar  must  be  brought  into  inti- 
mate contact.  To  make  this  adjustment,  insert  the  test  piece 
and  tighten  the  clamp  screws  of  the  cap.  Let  the  rocking  bar 
rest  on  the  lower  of  the  two  adjusting  screws,  and  adjust  this 
screw  till  the  contact  between  the  faced  surfaces  is  accurate. 
The  operator  should  look  across  the  contact  surface  toward  a 


XII,  §  230]         EXPERIMENTAL  METHODS  341 

strong  light.  A  well  lighted  window  is  the  best.  Unclamp 
the  upper  screws,  remove  the  test  bar,  and  set  W  on  the  zero 
mark,  adjusting  the  counterpoise  W  for  equilibrium. 

(2)  Having  demagnetized  the  bar,  insert  the  upper  end  in 
the  contact  piece  and  tighten  the  screws.     Hold  the  rocking 
bar  firmly  against  the  lower  adjusting  screw,  press  up  the  test 
bar,  clamp  into  contact  position,  and  tighten  the  lower  screws. 
Then  unscrew  the  lower  adjusting  screw  a  trifle,  perhaps  a 
quarter  of  a  turn,  to  insure  contact  between  the  faced  surfaces. 

(3)  Through  the  magnetizing  coil  C  pass  a  small  current 
sufficient  for  the  desired  minimum  value  of  H.     Then  slide  out 
the  weight  W  until  a  gentle  tap  on  the  base  with  the  finger 
causes  the  contact  faces  to  part,  and  record  the  position  of  the 
weight  on  the  graduated  bar.     This  is  the  value  of  D. 

(4)  Repeat  the  procedure  in  (3)  for  perhaps  ten  increasing 
values  of   the  current  strength  up   to  that  which  gives  the 
desired  maximum  value  of  H.     Let  each  reading  be  the  mean 
of  at  least  three  settings. 

(5)  Calculate  B  from  equation  (40),  remembering  that  W, 
the  weight  of  the  sliding  mass,  is  a  force,  and  must  be  ex- 
pressed in  dynes.     Corresponding  values  of  H  are  computed 
from  the  equation 

(«,  "-(>£> 

where  N  is  the  number  of  magnetizing  turns  and  L  is  the  free 
length  of  the  bar. 

(6)  Plot  the  B-H  curve  for  the  sample. 

The  utmost  care  is  necessary  to  avoid  injury  to  the  faced  surface  of  the 
cap.  This  piece  is  very  soft,  and  a  slight  blow,  or  a  fall  to  the  table  or 
floor,  or  careless  use  of  the  calipers  in  measuring  its  diameter  may 
seriously  impair  the  accuracy  of  the  facing.  Before  beginning  a  test  wipe 
both  faces  carefully  with  a  cloth. 

No  traction  method  can  be  regarded  as  very  satisfactory  for  investi- 
gating the  magnetic  quality  of  a  specimen.  Such  methods  are  at  best 
inexact  and  are  not  to  be  compared  with  ballistic  methods  for  accuracy. 


342 


MAGNETIC  TESTING 


[XII,  §  230 


They  afford  ready  means  for  making  approximate  and  rapid  tests, 
however,  and  they  are  especially  useful  for  comparative  methods  and 
for  grading,  where  results  are  desired  for  a  single  point  on  the  curve. 
A  joint  in  a  magnetic  circuit  always  increases  its  reluctance.  Between 
the  polished  faces  there  will  be  a  slight  adhesion,  tending  to  make  the 
values  of  B  too  great,  especially  at  low  values. 

The  advantages  of  the  method  are  that  it  is  simple,  that  the  test  pieces 
are  easily  prepared,  and  that  the  results  are  not  affected  by  stray  fields. 
Corrections  may  be  found  and  applied  for  the  air  gap  and  joint  reluctance, 
and  for  the  reluctance  of  the  yoke. 

231.  Air  Gap  Methods.  Let  S  be  a  bar  for  which  the  B-H 
curve  is  known,  and  let  T  be  another  bar  to  be  tested.  If  both 
bars  are  clamped  between  two  soft  iron  yokes,  as  shown  in 
Fig.  162,  the  number  of  magnetizing  turns  on  T  may  be  varied 
by  means  of  switches  until  the  flux  density  is  the  same  in  both 
bars.  When  this  condition  is  attained  there  will  be  no  leak- 
age between  points  A  and  B  of  the  yokes.  The  magnetic 
potential  of  the  points  is  the  same,  and  all  the  flux  lines  that 
pass  to  the  right  through  S,  pass  through  T  to  the  left.  If 
two  curved  horns  are  erected  at  the  points  A  and  B  as  shown 
in  Fig.  163,  a  magnetic  needle  in  the  air  gap  at  G  will  not 

be  deflected  when  the 


magnetizing  current  is 
reversed. 

This  method,  which 
is  due  to  Ewing,  is 
analogous  to  the  Wheat- 
stone  bridge  method  for  the  measurement  of  ohmic  resistance. 
While  it  is  not  now  in  general  use,  it  contains  the  essentials  of 
several  practical  methods. 

One  of  these  adaptations  of  Ewing's  method,  due  to  Koepsel, 
is  represented  in  Fig.  164.  The  test  bar  B  is  clamped  in  the 
pair  of  yokes  FF,,  and  magnetized  by  current  sent  through 
the  coils  mm'.  At  G  is  suspended  a  coil  similar  to  that  used 
in  a  d'Arsonval  galvanometer,  through  which  a  weak  current 


B 

0 

S 

T 

FIG.  162. 

XII,  I  231]        EXPERIMENTAL  METHODS 


343 


FIG.  103. 


of  constant   strength  is  maintained  by  an  auxiliary  battery. 
Any  magnetic  flux  set  up  in  the  air  gap  will  react  with  the 
field  about  the  coil,  and 
tend  to  rotate  the  sus- 
pended    system.      The 

apparatus  is  calibrated  /  A  (  \JB 

in  terms  of  a  standard  r- 
bar,  and  the  scale  ssf 
may  be  made  to  read 
directly  in  gausses.  An  ammeter  in  the  magnetizing  coil 
circuit  may  be  calibrated  in  C.  G.  S.  units  of  H.  The  method 
has  certain  defects,  due  to  the  reluctance  of  the  yoke  and  joints, 
and  to  the  field  which  the  coil  sets  up  outside  of  the  test  bar. 

In  an  improved  apparatus  of  the  same  general  design,  the 
pole  faces  at  G  (Fig.  164)  are  accurately  bored  out,  and  an 
armature  which  is  independently  driven  by  an  electric  motor 

is  inserted.  Therefore 
an  electromotive  force  is 
developed  if  there  is  any 
flux  across  the  gap,  and 
that  electromotive  force 
is  proportional  to  the  flux 
if  the  speed  of  rotation 
is  constant.  A  voltmeter 
connected  to  the  armature 
brushes  may  be  calibrated  to  read  either  flux  density  or  total 
flux.  The  ammeter  in  the  magnetizing  coil  circuit  may  be 
graduated  to  read  the  strength  of  the  magnetizing  field  in  any 
desired  units.  Full  correction  and  compensation  is  effected 
for  the  yoke  and  joint  reluctance,  the  independent  field  due  to 
the  coil,  and  leakage,  and  the  results  are  highly  satisfactory. 
The  apparatus  is  calibrated  by  means  of  a  standard  bar,  whose 
B-H  curve  has  been  determined  by  a  double  bar  and  yoke 
method  or  by  a  compensated  permeameter. 


f 

Y 

,  >    *  v 

5 

m 

B 

ooo.oooooooooooo 

Zl 

ooooooooooooooo 

B 

FIG.  164. 

344  MAGNETIC  TESTING  [XII,  §  231 

The  metal  bismuth  possesses  the  characteristic  of  under- 
going a  change  in  its  ohmic  resistance  when  placed  in  a  mag- 
netic field.  With  values  of  the  induction  density  varying  from 
zero  to  40,000  C.  G.  S.  units,  the  resistance  of  pure  bismuth 
wire  will  increase  nearly  threefold.  Advantage  is  taken  of 
this  property  for  measuring  the  magnetic  flux  through  an  air 
gap.  A  fine  bismuth  wire  is  wound  non-inductively  in  a  flat 
spiral  coil  and  incased  in  a  mica  capsule,  the  entire  thickness 
being  less  than  one  millimeter.  The  normal  resistance,  R,  of 
the  wire  having  been  measured,  it  is  placed  successively  in  fields 
of  increasing  strength,  the  resistance  being  measured  for  each 
value  of  H.  A  curve  showing  the  relation  between  values  of 
H  and  R  enables  us  to  interpolate  any  value  of  H  for  any  given 
value  of  R.  This  apparatus  is  sometimes  made  up  in  the  form 
of  a  Wheatstone  bridge,  the  scale  being  graduated  to  read  H 
units  directly. 

232.  Hysteresis.  Step-by-step  Method.  In  order  to  deter- 
mine the  contour  of  the  hysteresis  loop,  simultaneous  values 
of  H  and  B  must  be  determined  throughout  a  complete  mag- 
netic cycle.  A  convenient  method  for  doing  this  makes  use  of 
the  circuit  and  apparatus  described  in  §  223,  but  the  rheostat 
must  be  designed  so  that  definite  changes  may  be  made  in  the 
strength  of  the  current  without  breaking  the  circuit.  The 
magnetizing  field  H  is  increased  by  suitable  increments,  and 
the  corresponding  deflections  on  the  galvanometer  are  observed. 
Each  deflection  is  proportional  to  the  change  in  B  which  ac- 
companies the  change  in  H,  and  the  value  of  B  for  any  point 
is  found  by  adding  the  increments  in  B  from  the  beginning. 
The  separate  values  of  AB  may  be  calculated  from  equation 
(18),  and  added  together  ;  or  the  deflection  D  for  any  point 
may  be  regarded  as  the  sum  of  all  the  deflections  for  the  pre- 
ceding steps.  Any  value  of  B  is  a  AB  with  reference  to  the 
starting  point. 


XII,  §  233]         EXPERIMENTAL  METHODS 


345 


233.   Laboratory  Exercise  LIII.     To  determine  the  hysteresis 
in  a  sample  of  iron.     Step-by-step  method. 

APPARATUS.  As  in  Laboratory  Exercise  L,  §  223,  with 
a  special  rheostat.. 

PROCEDURE.  (1)  Arrange  the  circuit  as  for  the  ring 
method  (Fig.  154).  With  the  double  pole  double  throw 
switch  toward  the  M  side,  calibrate  the  ballistic  galvanometer 
as  usual. 

(2)  After  demagnetizing  the  ring,  close  the  battery  circuit 
with  Rf  set  for  a  small  current  value.  This  gives  a  magnetiz- 
ing force  of  H!,  and  a  throw  di  (Fig.  165).  Diminish  R'  by 
a  suitable  step  without  breaking  the  circuit.  This  gives  a 
magnetizing  force  H2  and  a  throw  d2.  Continue  in  this  way 
until  the  point  a  is  reached  in 
about  six  steps.  Rock  the  re- 
versing switch  about  twenty  times 
in  order  to  establish  this  point, 
meanwhile  having  the  key  k  open. 

Decrease  the  current  by  in- 
creasing R',  and  return  in  a  few 
steps  to  zero  current  at  point  b. 
Throw  over  the  reversing  switch 
and  increase  the  current  by  steps 
as  before  until  the  lower  maxi- 
mum is  reached  at  c.  The  current 
value,  and  hence  H,  should  be  the  same  as  at  point  a.  Estab- 
lish this  point  as  before  by  several  reversals. 

Return  along  cd  by  decreasing  the  current  until  d  is 
reached,  at  which  point  the  current  and  H  are  both  zero. 
Reverse  the  current  and  increase  by  steps  to  point  a.  The 
entire  cycle  should  have  about  twenty-five  steps.  The  number 
of  secondary  turns  may  be  changed  at  any  time  if  necessary, 
if  proper  account  is  taken  of  the  change  in  computing  B. 
Great  care  must  be  taken  in  observing  the  throws,  because 


FIG.  165. 


346 


MAGNETIC  TESTING 


[XII,  §  233 


any  error  in  one  value  of  d  will  carry  through  to  the  end 
when  the  deflections  are  summed. 

(3)  Calculate  values  of  H  from  equation  (21).  For  each 
value  of  H  the  corresponding  value  of  B  is  calculated  from 
equation  (18),  D  of  the  formula  being  the  sum  of  all  the  de- 
flections from  zero  up  to  that  point.  For  example,  the  first 
value  of  BX  corresponding  to  Hj  is  calculated  from  d^  The 
next  value  B2  is  calculated  from  di  +  d2,  and  so  on.  The  data 
and  results  may  be  tabulated  as  shown. 


B 


(4)  When  the  cusps  at  a  and  c  are  reached  the  deflection 
will  change  sign,  because  the  induction  changes  from  an  in- 
creasing to  a  decreasing  value.     In  view  of  this  change  it  is 
advisable  to  interchange  the  galvanometer  terminal  connec- 
tions at  these  points,  in  order  that  all  the  throws  may  be 
toward  the  same  side  of  the  scale.     This  obviates  any  error 
due  to  an  inequality  of  fiber  torsion.     The  values  of  d  must 
be  added  algebraically,  each  with  its  proper  sign. 

(5)  If  at  any  time  a  resistance  step  is  taken  so  great  that 
the  galvanometer  throw  is  off  the  scale,  the  previous  work 
must  be  discarded  and  the  entire  cycle  repeated.     In  order  to 
guard   against   this,   a   preliminary  cycle   should   be   carried 
through,  choosing  the  steps  so  that  the  throw  is  always  read- 
able.    The  numbers  of  the  steps  chosen  on  the  rheostat  should 
be  recorded  and  carefully  followed  in  the  subsequent  observing 
program. 

(6)  It  will  frequently  be  found  that  the  curve  as  drawn 


XII,  §  233]         EXPERIMENTAL  METHODS  347 

does  not  close  at  a.     It  is  probable  that  this  is  due  to  faulty 
readings  or  to  errors  in  summation  of  d. 

This  method  is  subject  to  the  criticism  that  the  throw  observed  is  not 
a  true  measure  of  the  change  in  B,  since  there  is  a  slow  creeping  up  of 
the  induction  which  may  persist  for  a  few  seconds  after  the  throw  has 
taken  place.  This  is  due  to  the  so-called  magnetic  viscosity.  A  mag- 
netometer method  is  free  from  this  error. 

(7)  The  antecedent  magnetic  state  is  of  great  significance 
in  all  magnetic  testing.     If  the  piece  has  residual  magnetism 
there  is  no  definite  relation  between  the  impressed  H  and  the 
induction  density   B.     If  the  sample   is   not   effectively   de- 
magnetized, the  rising  branch  of  the  loop,  da  (Fig.  165)  may 
cross  the  normal  induction  curve.     Moreover,  the  loop  will 
not  be  symmetrical  above  and  below  the  H  axis.     One  half 
of  the  difference  between  the  extreme  values  of  B,  however, 
gives  the  value  of  Bmax. 

Sometimes  the  branch  oa  is  omitted.  The  current  is 
brought  to  the  desired  maximum  value,  reversed  several  times 
and  decreased  by  steps,  the  starting  point  being  taken  at  a. 
The  values  of  H  and  B  will  then  be  plotted  with  reference  to 
a  temporary  pair  of  axes  as  shown  in  the  dotted  lines  in  Fig. 
165.  The  axes  through  o  may  be  drawn  subsequently,  if 
desired. 

(8)  Determine  the  area  of  the  loop  with  the  planimeter, 
calculate  the  energy  loss  in  ergs  per  cubic   centimeter   per 
cycle,  and  express  the  result  also  in  watts  per  pound  at  a 
frequency  of  60.     The  value  of  Bmax  should  always  be  stated 
in  connection  with  this  result. 

The  calculation  of  the  B  and  H  values  is  somewhat  laborious.  It  is 
advisable  to  make  some  trial  calculations  which  will  indicate  whether 
the  data  may  be  expected  to  yield  satisfactory  results.  Two  methods 
for  checking  the  correctness  of  the  data  are  given  in  the  following 
paragraphs. 

Beginning  at  o  (Fig.  165),  the  sum  of  the  deflections  from  o  to  a 
should  be  one  half  of  the  value  found  by  summing  the  deflections  from 
a  to  c.  Moreover,  the  summation  of  deflections  over  cda  should  be  equal 


348 


MAGNETIC  TESTING 


[XII,  §  233 


to  that  over  abc.     If  these  relations  are  not  found,  we  may  infer  that  the 
loop  will  not  close  at  a. 

Another  method  is  that  of  plotting  roughly  a  trial  curve  between  the 
sums  of  the  deflections  and  the  corresponding  current  values.  If  this 
curve  closes  and  has  a  proper  contour,  success  may  be  anticipated  with 
the  B-H  curve,  since  B  values  are  proportional  to  Zd,  and  H  values  are 
proportional  to  the  current. 

234.  Hysteresis.  Fixed  Point  Method.  It  has  been  seen 
that  the  area  of  the  hysteresis  loop  is  a  measure  of  the  energy 
loss  per  cycle,  when  iron  or  steel  is  subjected  to  a  cyclic 
magnetization.  The  determination  of  the  B  values,  as  out- 
lined in  the  step-by-step  method  of  §  232,  is  open  to  certain 
objections.  Errors  in  the  readings  are  cumulative,  and  the 
readings  for  individual  points  cannot  be  repeated.  The 


Bmax. 


-H 


H"H' 


H2 


FIG.  166. 

method  described  in  the  present  article  is  free  from  these 
faults,  inasmuch  as  each  reading  is  referred  to  a  fixed  point. 

Assume  any  chosen  magnetizing  force  H  applied  to  the  test 
piece,  which  brings  the  iron  to  some  point  Bma,  (Fig.  166). 
If  H  is  now  reduced  suddenly  to  a  value  H],  the  induction 
density  drops  to  Bb  and  the  change  in  B  which  causes  the 


XII,  §  234]         EXPERIMENTAL  METHODS  349 

galvanometer  throw  D  may  be  called  AXB.  The  relation  be- 
tween D  and  the  value  of  A  B  is  seen  in  equation  (18),  §  222. 
The  iron  is  now  brought  back  to  its  original  state  at  a  and  again 
H  is  reduced,  this  time  to  H2,  which  brings  the  iron  to  B2,  and 
the  corresponding  change  in  induction  is  A2B.  Continuing  in 
this  way  for  several  steps,  we  reach  the  point  b  on  the  curve. 

Points  from  b  to  c  may  be  found  by  so  arranging  the  appa- 
ratus that  the  current  corresponding  to  the  value  of  H  at  a  is, 
by  a  single  throw  of  the  switch,  (1)  reduced  to  zero,  (2)  re- 
versed, and  (3)  brought  to  some  small  initial  value  in  the 
opposite  direction.  This  impresses  on  the  iron  a  magnetizing 
force  H',  which  corresponds  to  an  induction  density  B'.  The 
change  in  induction  density  from  its  previous  value  may  be 
represented  by  A'B.  From  the  original  point  a,  proceeding 
in  the  same  manner,  we  reach  the  point  H",  with  the  corre- 
sponding induction  density  B".  In  this  way  successive  values 
of  H  are  established,  until  finally  the  point  c  is  reached,  which 
corresponds  to  a  reversal  of  the  original  value  of  the  mag- 
netizing current. 

The  galvanometer  throw  observed  when  the  iron  is  carried 
from  the  point  a  to  the  point  c  is  due  to  a  change  in  induction 
density  given  by  the  equation 


The  value  of  Bmax  is  taken  as  one  half  of  the  total  change  in 
B,  corresponding  to  a  complete  reversal  of  the  original  mag- 
netizing current,  and  other  values  of  B  may  be  found  by  sub- 
tracting in  succession  the  various  values  of  AB  from  Bmax. 
The  values  of  H  may  be  calculated  from  equation  (21).  With 
these  corresponding  values  of  H  and  B  the  curve  from  a 
through  6  to  c  may  be  plotted.  The  other  half  of  the  loop 
may  be  drawn  in  from  symmetry.  Further  details  of  the 
method  will  be  given  in  connection  with  the  Laboratory  Exer- 
cise LIY,  §  235. 


350 


MAGNETIC  TESTING 


[XII,  §  235 


235.  Laboratory  Exercise  LIV  :  To  determine  the  hysteresis 
loss  in  a  sample  of  iron  by  the  fixed  point  method. 

APPARATUS.  The  same  as  in  Laboratory  Exercise  L,  §  223, 
with  an  additional  rheostat  and  a  knife  switch. 

PROCEDURE.  (1)  Connect  the  circuit  as  in  Fig.  167.  The 
reversing  switch  W  is  so  arranged  that  the  points  c  and  d  are 
connected  in  the  usual  manner,  while  the  points  a  and  /  are 
connected  through  an  adjustable  resistance  R2.  The  knife 


Fia.  167. 

switch  or  key  at  K  is  used  to  short-circuit  this  resistance  if 
desired.  An  examination  of  the  figure  will  show  that  the 
switch  W  is  an  ordinary  reversing  switch  when  K  is  closed. 
When  K  is  open,  and  W  is  on  the  points  ad,  the  current  has  a 
maximum  value  determined  by  R^  When  W  is  on  the  points 
c/,  the  current  will  be  diminished  by  the  introduction  of  R2  if 
IT  is  open. 

(2)  With  the  double  pole  double  throw  switch  T  thrown  to 
the  mutual  inductance  side  of  the  circuit,  and  with  the  key  K 
closed,  calibrate  the  galvanometer  in  the  usual  way. 


XII,  §  235]         EXPERIMENTAL  METHODS  351 

(3)  With  the  switch  T  on  the  ring  side  of  the  circuit  and 
with  Won  the  points  c/,  in  which  case  the  cross  connections 
are  in  the  magnetizing  current  circuit,  adjust  ^  until  a  current 
value  is  reached  such  that  the  B  limits  will  be  about  6000 
or  8000  gausses.     It  is  left  as  an  exercise  for  the  student  to 
show  how  the  current  is  determined  which  corresponds  to  this 
value  of  B^.     With  the  key  k  open,  and  K  closed,  rock  the 
switch  back  and  forth  several  times  in  order  to  bring  the  iron 
into  a  cyclic  state,  finally  resting  on  the  points  c/. 

(4)  To  obtain  points  on  the  curve  ab  (Fig.  166),  between  the 
maximum  positive  value  of   H  and  zero,  proceed  as  follows. 
With  the  switch  W  on  the  points  cf,  and  with  a  small  resistance 
in  jR2J  quickly  open  K  and  read  the  throw  due  to  the  sudden 
decrease  in  the  induction  density  AjB.     Close  K  and  rock  W 
several  times  in  order  to  establish  again  the  original  point  a, 
with  k  open  to  avoid  damaging  the  galvanometer.     Increase 
R2,  open  K  again,  and  read  the  throw  as  before ;  this  reading 
corresponds  to  a  change  A2B.     Continue  in  this  way  for  about 
six   steps   until   the  magnetizing   current   has   been  reduced 
to  zero. 

(5)  Points  on  the  curve  from  b  to  c  now  remain  to  be  deter- 
mined.    With   the   original   value   of   the   current,   rock   the 
reversing  switch    W  several  times  as  before  with  K  closed, 
resting  finally  on  points  ad1,  in  which  case  the  cross  connections 
are  not  in  the  current  circuit.     The  iron  is  now  at  the  point  c 
(Fig.  166) ;  to  regain  the  value  of  B^  at  a,  the  current  through 
the  magnetizing  turns  of  the  ring  must  be  reversed.     This  may 
be  done  conveniently  by  interchanging  the  current  wires  at  the 
primary  terminals  of  the  ring. 

Eeversing  the  switch  W  now  will  bring  the  original  value 
of  the  magnetizing  current  to  zero  and  reestablish  it  in  the 
opposite  direction.  If  at  the  same  time  the  resistance  Rz  is 
introduced  into  the  circuit,  the  limit  of  the  magnetizing  field  in 
the  negative  direction  is  fixed  at  some  value  less  in  magnitude 


352  MAGNETIC   TESTING  [XII,  §  235 

than  the  preceding  positive  value.  With  K  open,  set  R^  at 
some  high  value  which  ensures  a  suitable  small  negative  incre- 
ment H',  and  read  the  throw  when  W  is  thrown  over.  This 
corresponds  to  some  change  in  the  induction  density  A'B,  and 
the  iron  is  now  at  the  point  B'. 

Again  establish  the  point  a  by  repeated  reversals  with  K 
closed.  With  H^  slightly  decreased  and  with  A"  open,  reverse 
the  switch  and  read  the  throw  corresponding  to  a  change  A"B, 
from  Bmax  to  B".  Proceed  in  this  way  until  the  point  c  is 
reached. 

The  last  step  is  taken  with  K  closed,  which  corresponds  to  a 
complete  reversal  of  the  magnetizing  field  H,  and  gives  the 
iron  an  induction  density  of  —  Bmax. 

Throughout  the  foregoing  procedure  the  rheostat  Rly  has  not 
been  changed,  and  the  current  should  return  to  its  original  value. 

(6)  In  order  to  prevent  dangerous  throws  of  the  galvanom- 
eter the  key  k  should  be  kept  open  except  when  a  reading  is  to 
be  made.     Care  should  be  taken  to  rock  the  reversing  switch 
W  several   times  .before   each   reading  in  order  to  bring  the 
induction  density  to  a  point  which  corresponds  to  the  maxi- 
mum value  of  H. 

(7)  This  method  possesses  a  distinct   advantage  over  the 
step-by-step  method,  in  that  each  point  on  the  curve  is  deter- 
mined independently,  and  is  reached  by  a  single  step  from  the 
end  of  the  cycle.     This  prevents  the  carrying  forward  and  accu- 
mulation of  errors.     Each  reading  may  be  repeated  as  many 
times  as  desired,  and  its  correctness  checked.     The  cusps  of 
the  loop  fall  on  the  normal  induction  curve. 

(8)  Calculate  values  of  AB  and  H  from  equations  (18)  and 
(21).     Beginning  at  point  a,  plot  the  portion  of  the  curve  abc 
from  the  values  of  A  B  and  H  as  computed  above.     Draw  in  the 
other  side  of  the  loop  cda  by  symmetry  and  locate  the  perma- 
nent axes  through  o.     Enter  on  these  axes  the  appropriate 
scales  of  B  and  H  values.     On  the  curve  the  value  of  Bmax  is 


XII,  §  236] 


EXPERIMENTAL  METHODS 


353 


represented  by   one   half   of   the    extreme   vertical   distance 
between  the  cusps. 

(9)  Determine  the  area  of  the  completed  loop  with  a  planim- 
eter,  and  calculate  the  energy  loss  in  ergs   per  cubic  centi- 
meter per  cycle.     Also  convert  this  into  the  equivalent  watts 
per  pound  at  a  frequency  of  60  cycles  per  second. 

(10)  Eepeat  the  experiment,  using  a  magnetizing  current 
sufficient  to  produce  B  limits  of  10,000  gausses. 

236.  Laboratory  Exercise  LV:  To  determine  the  flux  density 
in  a  permanent  magnet. 

APPARATUS.  Standard  mutual  inductance,  ballistic  gal- 
vanometer, slip  coil  and  magnet  to  be  tested,  battery,  re- 
versing switch,  and  ammeter. 

PROCEDURE.  (1)  Arrange  the  circuit  as  in  Fig.  168.  Con- 
nect the  slip  coil  in  series  with  the  ballistic  galvanometer  and 
the  secondary  of  the  mutual  inductance.  Place  the  slip  coil 
over  the  middle  of  the  bar  magnet  so  that  the  maximum 
number  of  lines  of  force  link  with  it. 
Withdraw  the  coil  quickly  and  read 
the  throw  c?x.  The  quantity  induced 
in  the  galvanometer  circuit  is  given 
by  the  formula 


(42)     Ql  =         = 


where  ABJ.  is  the  total  number  of 
flux  lines  linked  with  the  coil.  In 
this  equation,  A N  is  the  change  in 
the  number  of  liiikings,  AB  is  the  FIG.  168. 

change  in  the  induction  density  (from 

B  to  zero),  JS  is  the  number  of  wire  turns  in  the  coil,  A  is  the 
cross  section  of  the  bar,  and  R  is  the  total  secondary  circuit 
resistance. 
2  A 


354  MAGNETIC  TESTING  [XII,  §  236 

(2)  Adjust  R'  so  that  a  suitable  current  of  strength  i  passes 
through  the  primary  coil  of  the  mutual  inductance  M,  and 
reverse  the  switch,  reading  the  throw  d2.  The  quantity  in- 
duced in  the  galvanometer  circuit  is 

(43)  Q, 

Dividing  (42)  by  (43),  and  solving  for  AB  we  have 


It  is  evident  in  this  equation  that  AB  is  the  value  of  B  in 
the  bar. 

If  Jf,  i,  and  A  are  in  absolute  C.  G.  S.  units,  then  B  will  be 
given  in  gausses.  If  M  is  in  henrys  and  i  is  in  amperes,  the 
equation  for  B  becomes 

(45)  B  =  ^|H, 

(3)  Repeat  the  readings  for  dj  and  d2  several  times  and 
calculate  the  value  of  B. 

(4)  State  in  connection  with  the  value  of  B  the  dimensions 
and  shape  of  the  test  piece,  and  calculate  the  value  of  the  in- 
tensity of  magnetization. 

It  must  be  remembered  that  this  test  gives  information  for 
this  particular  form  of  the  test  piece,  and  yields  results  char- 
acteristic of  the  material  used  only  when  the  bar  is  very  long. 

237.  The  Measurement  of  Core-Loss.  Hysteresis  has 
been  defined  as  the  lag  of  the  change  in  induction  density 
behind  the  change  in  the  magnetizing  force.  In  order  to  carry 
a  piece  of  iron  through  a  complete  magnetic  cycle  a  certain 
amount  of  energy  is  required.  This  ultimately  takes  the  form 
of  heat,  and  is  known  as  the  hysteresis  loss.  Moreover,  on 
account  of  the  cyclic  magnetization,  eddy  currents  are  set  up 
in  the  iron,  and  these  also  represent  a  definite  energy  equivalent. 


XII,  §  237]         EXPERIMENTAL  METHODS  355 

In  iron  cores  used  with  alternating  current  circuits  these  effects 
are  closely  associated,  and  can  only  be  determined  separately 
by  careful  measurements.  These  combined  effects  are  known 
as  core-loss,  and  this  may  be  expressed  in  terms  of  energy 
units  per  unit  mass  of  material  per  cycle,  or  better,  it  may  be 
expressed  by  the  power  consumption  in  watts  per  pound  or 
per  kilogram  of  material,  for  a  given  frequency  and  a  given 
maximum  induction  density. 

A  method  due  to  Epstein  is  widely  used  for  making  core- 
loss  tests.  The  following  paragraphs  which  describe  this 
method  are  adapted  from  the  standard  specifications  for 
magnetic  tests  as  adopted  by  the  American  Society  for  Testing 
Materials  in  19111. 

The  standard  core-loss  is  the  power  in  watts  consumed  in 
each  kilogram  of  material  at  a  temperature  of  25°  C.,  when 
submitted  to  a  harmonically  varying  induction,  having  a 
maximum  of  10,000  gausses  and  a  frequency  of  60  cycles  per 
second,  when  measured  as  specified. 

The  magnetic  circuit  consists  of  10  kilograms  (22  pounds)  of 
the  test  material,  cut  into  strips  50  centimeters  long  and  3 
centimeters  wide,  half  parallel  and  half  at  right  angles  to  the 
direction  of  rolling.  These  are  assembled  in  four  equal  bundles 
and  arranged  in  the  four  sides  of  a  square  with  butt  joints,  the 
opposite  sides  consisting  of  strips  cut  from  the  sheets  in  the 
same  way  with  reference  to  the  direction  of  rolling. 

No  insulation  other  than  the  natural  scale  of  the  material 
(except  in  the  case  of  scale  free  material)  is  used  between 
laminations,  and  the  corner  joints  are  separated  by  paper 
strips,  0.01  inch  in  thickness. 

The  magnetizing  winding  consists  of  four  solenoids  surround- 
ing the  four  sides  of  the  magnetic  circuit,  and  joined  in  series. 
This  is  represented  in  Fig.  169  as  a  single  coil  M.  A  secondary 

1  Transactions  of  the  American  Society  for  Testing  Materials,  1911 ;  Vol. 
XI,  p.  110. 


356 


MAGNETIC  TESTING 


[XII,  §  237 


coil  P  is  used  for  energizing  the  voltmeter  and  the  potential 
coil  of  the  wattmeter.  These  coils  are  wound  on  forms  of 
non-magnetic,  non-conducting  material,  of  the  following 
dimensions : 


Inside  cross  section : 
Thickness  of  wall : 
Winding  length : 


4  by  4  centimeters, 
not  over  0.3  centimeter. 
42  centimeters. 


The  winding  on  each  solenoid  should  consist  of  150  turns  of 
copper  wire  uniformly  wound  over  the  42  centimeters  of  length. 
The  total  resistance  of  the  magnetizing  coils  is  between  0.3 
and  0.5  of  an  ohm.  The  secondary  winding  of  150  turns  of 
copper  wire  on  each  solenoid  is  similarly  wound  beneath  the 
primary  coils.  Its  resistance  should  not  exceed  one  ohm. 


A.  C,  mains 


FIG.  169. 

The  arrangement  of  the  .circuit  is  shown  in  Fig.  169.  A 
voltmeter  V,  and  the  potential  coil  of  a  wattmeter  W,  are  con- 
nected in  parallel  to  the  terminals  of  the  secondary  winding  of 
the  apparatus.  The  current  coil  of  the  wattmeter  is  connected 
in  series  with  the  primary  winding.  A  frequency  meter  is 
placed  at  F.  An  alternating  E.  M.  E.  of  the  pure  sine  curve 
typQ  is  impressed  on  the  primary  windings,  and  is  adjusted 


XII,  §  237]         EXPERIMENTAL   METHODS  357 

until  the  voltage  of  the  secondary  circuit  is  given   by   the 
equation 


where  the  letters  denote  the  following  quantities  : 

/=  Form  factor1  of  the  primary  E.  M.  F.  wave  =  1.11  for  a 
sine  wave. 

N  =  Number  of  secondary  turns  =  600. 

n    =  Number  of  cycles  per  second  =  60. 

B  =  Maximum  induction  density  ==  10,000. 

M  =  Total  mass  in  grams  =  10,000. 

L  =  Length  of  strips  in  centimeters  =  50. 

D  =  Specific  gravity. 

E  =  106.6  volts  for  a  sine  wave  with  high  resistance  steel. 

E  =  103.8  volts  for  a  sine  wave  with  low  resistance  steel. 

A  specific  gravity  of  7.5  is  assumed  for  all  steel  samples  having 
a  resistivity  of  more  than  two  ohms  per  meter-gram,  and  7.7 
for  all  steel  samples  having  a  resistivity  of  less  than  two  ohms 
per  meter-gram.  These  are  designated  as  high  resistance 
and  low  resistance  steels  respectively.  The  formula  is  derived 
as  follows. 

The  maximum  magnetic  flux  through  the  iron  or  steel  may 
be  represented  by  <f>.  This  will  rise  and  fall  and  reverse  its 
direction  with  the  magnetizing  field  of  the  alternating  current, 
and  it  will  cut  the  wire  turns  of  the  voltage  coil  four  times  per 

1  If  the  average  is  taken  of  a  large  number  of  equidistant  ordinates  for  a 
sine  curve,  it  is  found  to  be  0.6369  of  the  maximum  ordinate.  That  is,  the 
average  value  of  the  E.M.F.  is  0.6369  of  the  maximum  value.  The  reading  on 
the  voltmeter,  however,  is  not  the  maximum  value,  nor  is  it  the  average 
value,  but  it  is  that  value  of  the  alternating  voltage  which  is  equivalent  to 
the  square  root  of  the  mean  of  the  squares  of  the  several  ordinates.  This  is 
called  the  effective  value,  and  may  be  shown  to  be  0.707  of  the  maximum 
ordinate.  The  ratio  of  the  effective  to  the  average  value  is  1.11  for  a  sine 
curve,  and  this  ratio  is  called  the  form  factor  of  the  wave.  Proofs  of  these 
relations  may  be  found  in  FLEMING,  The  Alternating  Current  Transformer, 
Vol.  1,  pp.  98-103. 


358  MAGNETIC  TESTING  [XII,  §  237 

cycle.  For  a  frequency  n  the  flux  will  cut  the  wire  turns  4  n 
times  per  second.  If  there  are  N  wire  turns  on  the  voltage 
coil  there  will  be  4  nN  flux  turns  or  linkings  per  second. 
From  the  Faraday  equation,  §  129,  the  average  voltage  induced 
is  equal  to  the  time  rate  of  change  of  the  number  of  linkings  ; 
whence  we  have 

(47)  tf^i-jg-a  volts. 

The  value  of  <^mai  may  be  replaced  by  ^Bmoar,  where  A  is  the 
area  of  cross  section  of  the  core.-  Moreover,  since  we  know 
that 

volume  =    mass    , 
density 

we  may  write 


or 

A         M 


(48)  A 
It  follows  that 

(49)  <t>m«x=BmaxM- 

Combining  equations  (47)  and  (49),  we  obtain  the  equation 

(50)  Ea 


Since  form  factor  is  defined  as  the  ratio  of  the  effective  to  the 
average  value  of  the  voltage  during  a  half  cycle,  equation  (50) 
may  be  written  in  the  form 


(51)  E     -E    f- 

1LD10*    ' 

whence 

fW\  R          4  LDEefJ  1Q8 

-          *» 


XII,  §  237]         EXPERIMENTAL  METHODS  359 

The  wattmeter  gives  the  power  consumed  in  the  iron  and 
the  secondary  circuit.  Subtracting  the  correction  terms  from 
the  total,  and  dividing  by  the  mass  in  kilograms,  gives  the 
loss  in  watts  per  kilogram  under  standard  conditions. 

It  is  often  desirable  to  separate  the  losses  due  respectively 
to  hysteresis  and  to  eddy  currents.  This  may  be  done  by 
taking  advantage  of  the  fact  that  for  a  given  value  of  the 
maximum  induction  density,  hysteresis  varies  directly  with 
the  frequency,  while  the  eddy  current  loss  varies  with  the 
square  of  the  frequency.  If  the  total  core-loss  is  determined 
for  two  known  and  different  frequencies  for  the  same  value  of 
Bmax)  two  simultaneous  equations  may  be  written  with  two  un- 
known quantities,  from  which  either  one  of  the  two  factors 
may  be  found. 

The  method  described  above  requires  expensive  equipment 
and  massive  samples,  and  it  may  be  replaced  oftentimes  by 
the  following  method,  which  yields  results  of  sufficient  accu- 
racy for  many  purposes.  A  small  bundle  of  strips  of  the  test 
material  is  placed  in  a  solenoid  energized  by  alternating  cur- 
rent, with  the  current  coils  of  the  wattmeter  in  series  with  the 
magnetizing  circuit.  Two  separate  secondary  windings  are 
provided  for  the  potential  coils  of  the  wattmeter  and  for  the 
voltmeter.  A  low  frequency  is  used  in  order  to  minimize  the 
eddy  current  effects.  Empirical  corrections  are  applied  for 
the  irregular  flux  distribution  in  the  sample.  Measurements 
thus  made  are  accurate  within  5  %. 

A  still  further  simplification  gives  results  sufficient  for 
comparative  purposes,  in  which  a  bundle  of  strips  of  the  test 
material,  of  specified  dimensions,  is  placed  in  a  magnetizing 
solenoid,  and  wattmeter  readings  are  compared  with  others 
taken  for  a  standard  sample  under  the  same  conditions. 

The  various  electrical  handbooks  give  valuable  information 
on  these  methods  of  testing,  and  also  on  the  values  of  core- 
loss  in  representative  materials. 


360  MAGNETIC  TESTING  [XII,  §  238 

238.  Laboratory  Exercise  LVI :  To  determine  core-loss  with 
the  Epstein  equipment. 

APPARATUS.  Epstein  permeaineter  and  accessories,  source 
of  suitable  alternating  current,  voltmeter,  ammeter,  wattmeter, 
frequency  meter,  and  samples  to  be  tested. 

PROCEDURE.  (1)  Arrange  the  circuit  as  shown  in  Fig.  169. 
Place  the  bundles  of  steel  in  the  solenoids,  with  strips  cut  the 
same  way  in  opposite  sides,  and  with  one  thickness  of  paper 
separating  the  ends.  Clamp  the  bundles  in  place  before  clos- 
ing the  line  switch.  Rap  the  joints  with  a  wooden  or  raw- 
hide mallet  until  the  ammeter  shows  a  minimum  magnetiz- 
ing current.  Short-circuit  the  ammeter  with  a  knife-switch. 
If  standard  conditions  are  to  be  used,  adjust  the  voltage  and 
frequency  to  normal.  Never  adjust  the  voltage  by  inserting 
inductive  resistance,  since  this  will  change  the  wave  form. 

(2)  Read  the  wattmeter  and  voltmeter  simultaneously  and 
note  the  frequency.     Subtract  from  the  wattmeter  reading  the 
E^/R  loss  in  the  potential  coils  of  the  two  instruments.     The 
losses  in  coils  of  the  permeameter  are  negligible. 

(3)  Divide  the  corrected  loss  by  the  mass.     This  gives  the 
core-loss  in  watts  per  kilogram.     Express  this  result  also  in 
watts  per  pound. 

(4)  Calculate  B  from  equation  (52). 

(5)  Repeat  for  three  other  samples. 

(6)  In  order  to  separate  the  hysteresis  effects  from  the  eddy 
current  loss,  the  test  may  be  repeated  at  a  different  frequency. 

EXERCISES 

1.  Reduce  a  loss  of  0.9  watt  per  pound,  Bmax  -  10,000,  at  100  cycles 
per  second,  to  ergs  per  cubic  centimeter  per  cycle. 

2.  Given  an  iron   ring  40  centimeters  in  diameter,  of  cross  section  6 
square  centimeters,  overwound  with  200  primary  turns,  and  100  secondary 
turns.     A  current  is  passed  through  the  primary  or  such  value  that  the 
iron  has  a  permeability  of  1000.     Calculate  the  mutual  inductance  of  the 
coils  in  millihenrys. 

3.  Show  that  the  number  of  ampere  turns  necessary  to  establish  a  flux 
density  B  through  an  air  gap  of  length  L  cm.  is  0.8  BZ». 


APPENDIX 

PABT  I.    ABSOLUTE  MEASUREMENTS 

239.  Absolute   Measurements.      An   important   class    of 
electric  measurements,  which  can  only  be  undertaken  in  a  well 
equipped  precision  laboratory,  includes  the  determination  of 
the  fundamental  electric   quantities,    current,   resistance   and 
potential  difference,  directly  in  terms  of  the  centimeter,  the 
gram,  and  the  second.1 

Bodies  which  are  electrically  charged  are  known  to  attract 
or  repel  one  another  with  forces  which  are  proportional  to 
the  magnitudes  of  the  charges.  An  instrument  called  the 
electrometer  is  used  to  measure  directly  the  force  action 
between  parallel  plates,  when  they  are  charged  with  a  given 
potential  difference.  The  theory  and  description  of  this  in- 
strument in  its  various  forms  will  be  found  in  the  larger  text- 
books on  electricity.  In  brief,  the  determination  of  a  poten- 
tial difference  is  made  to  depend  upon  force  measurements 
between  parallel  charged  plates.  On  account  of  the  difficulty 
of  making  plates  perfectly  plane,  and  keeping  them  exactly 
parallel,  the  precision  attained  is  not  high,  and  the  direct 
determination  of  a  potential  difference  in  absolute  measure  is 
not  attempted. 

The  International  Conference,  London,  1908,  selected  the 
ampere  and  the  ohm  as  the  two  units  which  could  be  most 
easily  evaluated  in  absolute  measure,  and  from  these  two,  the 
volt  is  readily  fixed  by  the  relation  given  in  Ohm's  Law. 

240.  The  Absolute   Measure  of  Current  Strength.    The 

forces  due  to  the  magnetic  fields  about  current-carrying  con- 

1  See  GRAY,  Absolute  Measurements  in  Electricity  and  Magnetism,  vol.  IT. 

361 


362  APPENDIX  [§  240 

ductors  may  be  measured  directly  in  dynamic  units  in  any 
one  of  the  three  following  ways. 

I.  If  a  standard  magnetic  field  is  available,  equations  sim- 
ilar to  those  for  the  tangent  galvanometer  will  give  the  value 
of  a  current  strength  in  terms  of  the  field  strength,  together 
with  certain  geometric  constants  of  the  circuit. 

II.  The  torque  due  to  the  reaction  of  magnetic  fields  about 
fixed  and  movable   coils  may  be  expressed  in  terms  of  the 
elastic  constants  of  a  suspension  fiber,  when  the  suspended 
system  moves  about  a  vertical  axis. 

III.  The  force  or  torque  due  to  the  magnetic  field  reactions 
about  adjacent  coils  may  be  balanced  against  the  gravitational 
pull  of  the  earth  011  known  masses,  if  the  axis  of  rotation  of 
the  system  is  horizontal. 

Absolute  methods  of  measurement  of  current  strength  have 
been  based  upon  all  of  these  principles,  but  it  is  by  means  of 


^s: 


FIG.  170. 

the  third  that  the  most  recent  and  the  most  accurate  deter- 
minations have  been  made.  The  principle  is  illustrated  by 
Fig.  170. 

The  coils  A,  B,  C,  and  D  are  fixed  in  position,  and  coils  E 
and  F  are  hung  symmetrically  between  them,  from  the  arms 
of  a  balance.  If  the  same  current  passes  in  series  through  all 
six  of  the  coils,  in  such  direction  that  the  acting  torques  are  all 
in  the  same  sense  with  respect  to  the  axis  of  rotation  0,  the 
tendency  of  the  beam  to  turn  may  be  compensated  by  the  pull 
of  the  earth  on  standard  masses  placed  on  the  beam  or  on  pans 


§  241]  ABSOLUTE   MEASUREMENTS  363 

attached  to  the  coils  E  and  F.  The  total  torque  is  proportional 
to  the  square  of  the  current  strength,  and  the  value  of  the 
current  may  be  calculated  from  the  forces  acting,  the  dimen- 
sions of  the  coils,  and  the  number  of  wire  turns.  In  recent 
work  the  result  is  made  to  depend  upon  the  ratio  of  the  radii 
of  the  fixed  and  movable  coils.  This  ratio  can  be  obtained 
by  an  electric  method  with  greater  precision  than  by  direct 
measurement. 

The  current  strength  determined  in  this  way  is  expressed  in 
absolute  C.  G-.  S.  units.  A  silver  voltameter  in  series  with  the 
coils  enables  the  experimenter  to  express  the  result  in  terms 
of  the  mass  of  silver  deposited  in  one  second.  Moreover,  if  a 
standard  resistance  is  in  series  with  the  circuit,  the  potential 
difference  across  its  terminals  when  the  measured  current  is 
flowing,  may  be  compared  with  a  Weston  normal  cell.  In  this 
way  the  value  of  the  International  ampere  may  be  expressed 
in  terms  of  absolute  units. 

The  precision  attainable  in  these  measurements  is  of  the 
order  of  two  or  three  parts  in  lOOjOOO.1 

241.  The  Absolute    Measurement   of    Resistance.     The 

necessity  of  finding  the  resistance  of  some  particular  conduc- 
tor in  absolute  measure  was  recognized  early.  In  1863  such 
determinations  were  undertaken  by  a  committee  of  the  Brit- 
ish Association.  Many  methods  have  since  been  suggested 
and  developed.  That  due  to  Lorenz  is  probably  the  simplest 
in  theory  and  in  practice. 

In  §  109  it  was  shown  that  a  standard  magnetic  field  of 
known  strength  is  realized  at  the  center  of  a  long  solenoid 
which  carries  a  steady  current.  Let  the  outer  circle  AA,  Pig. 
171,  represent  one  turn  at  the  middle  of  such  a  solenoid.  The 
magnetic  field  is  here  perpendicular  to  the  plane  of  the  paper. 

1  For  a  detailed  discussion  of  these  methods,  together  with  references  to 
standard  papers  on  the  subject,  see  U.S.  BUREAU  OF  STANDARDS,  Bulletin, 
Vol.  8,  p.  269,  1912. 


364 


APPENDIX 


[§241 


With  the  battery  polarity  as  indicated,  the  direction  of  the 
field  is  away  from  the  reader.  A  circular  disk  D  of  radius  r 
is  mounted  with  its  plane  at  right  angles  to  the  axis  of  the 
solenoid.  This  disk  can  be  maintained  in  uniform  rotation  by 

an  auxiliary  motor  of  any  kind, 
and  it  is  equipped  with  a  de- 
vice for  accurately  counting 
the  number  of  revolutions  per 
second.  Brush  contacts  are 
applied  at  the  axis  and  rim  of 
this  disk.  The  radial  filament 
of  the  disk  connecting  the  points 
of  contact  of  the  brushes  may  be 
regarded  as  a  conductor  which 
is  cutting  lines  of  force.  A 

constant  potential  difference  exists  between  the  brushes  when 
a  uniform  rotation  is  maintained.  Let  H  represent  the  flux 
density  through  the  disk  and  A  its  effective  area.  The  total 
flux  through  the  disk  is  then  given  by 


FIG.  171. 


and  this  flux  is  cut  once  by  the  line  connecting  the  brushes, 
for  each  complete  revolution.  If  the  time  of  one  revolution 
is  t  seconds,  then  the  number  of  lines  of  magnetic  flux  cut  in 
one  second  is  given  by  <£/£.  This  may  be  regarded  as  the  rate 
of  change  of  linking  of  magnetic  lines  with  the  radius  of  the 
disk  which  connects  the  brushes,  and  the  potential  difference 
between  the  contact  points  may  then  be  found  by  means  of 
Faraday's  equation  (7),  §  129. 
We  may  then  write 


dt       t        t 
By  equation  (33),  §  109,  we  have 

H  =  4  Trni 


§  241]  ABSOLUTE  MEASUREMENTS  365 

whence 


t 

From  the  equations  of  §  131,  it  is  seen  that  the  mutual  induc- 
tance between  the  disk  and  the  solenoid,  or  the  number  of  flux 
lines  passing  through  the  disk  when  unit  current  flows  in  the 
coil,  is  given  by  the  equation 

M  =  4  TrnA, 
whence 

E  =  —  - 

t 

The  brushes  are  connected  to  the  terminals  of  the  resistance 
R  which  is  to  be  determined,  and  a  galvanometer  g  is  included 
in  the  circuit.  The  connections  are  made  so  that  the  potential 
difference  due  to  the  cutting  of  magnetic  lines  is  opposed  to 
the  potential  drop  through  R.  If  the  value  of  i  and  the  speed 
of  rotation  are  adjusted  so  that  no  deflection  occurs  on  the 
galvanometer,  then  we  may  write 

—  =  iR 
t 

or 


Since  the  time  of  one  revolution  is  always  the  reciprocal  of 
the  number  of  revolutions  per  second  2,  we  have 


The  value  of  M  can  be  calculated  directly  from  the  geometric 
constants  of  the  apparatus,  and  z  can  be  determined  accurately. 
In  this  way  the  resistance  of  a  wire  coil  or  of  a  mercury  col- 
umn can  be  directly  evaluated  in  absolute  measure.1 

1  For  a  detailed  account  of  the  construction  of  primary  mercurial  resist- 
ance standards,  see  U.  S.  BUREAU  OF  STANDARDS,  Bulletin,  Vol.  12,  p.  375. 


366 


APPENDIX 


PART  II.     TABLE 

CONDUCTIVITY  OF  NORMAL  SOLUTION  OF  KC1 
[74.60  grams  of  KCI  in  one  liter  of  solution.] 


TEMP. 

CONDUCTIVITY 

TEMP. 

CONDUCTIVITY 

15°  C. 

0.09264  mhos 

22°  C. 

0.10595  mhos 

16 

0.09443 

23 

0.10788 

17 

0.09663 

24 

0.10981 

18 

0.09824 

25 

0.11174 

19 

0.10016 

26 

0.11392 

20 

0.10209 

27 

0.11614 

21 

0.10402 

28 

0.11840 

MATHEMATICAL  AND  PHYSICAL  CONSTANTS  IN  FREQUENT  USE 


Base  of  Napierian  logarithms 

To  convert  common  logarithms  into  Napierian,  mul- 
tiply by 

To  convert  Napierian  logarithms  into  common  loga- 
rithms, multiply  by 

Eatio  of  circumference  to  diameter  of  a  circle 

IT* 
47T 

J_ 

47T 
lOg  4  7T 

Density  of  mercury 
Electrochemical  equivalent  of  silver 

Acceleration  of  gravity,  Potsdam 
Acceleration  of  gravity,  Washington,  D.C. 


=     2.71828 
=     2.30258 

0.43429 
3.1416 


:  12.566 
:     0.07957 

:     1.0992 

:   13.5956  gr.  per 

cubic  centimeter 

:     0.00111804  gr. 

per  coulomb 

: 981. 274  cm.  sec. 

:  980. 094  cm.  sec. 


REFERENCE  BOOKS  367 


PART  III.     STANDARD  REFERENCE  BOOKS 

Fleming  —  Handbook  for  the  Electrical  Laboratory,  2  vols. 

Fleming  —  Alternate  Current  Transformer,  Vol.  I. 

Karapetoff — Experimental  Electrical  Engineering,  2  vols. 

Kempe  —  Handbook  of  Electrical  Testing. 

Kohlrausch  —  Lehrbuch  der  Praktischen  Physik. 

Roller — Electric  and  Magnetic  Measurements. 

Edgcumbe  —  Industrial  Electrical  Measuring  Instruments. 

Murdock  and  Ochswold  —  Electrical  Instruments. 

Ewing  —  Magnetic  Induction  in  Iron  and  other  Metals. 

Du  Bois  —  The  Magnetic  Circuit  in  Theory  and  Practice. 

Northrup  —  Methods  of  Measuring  Resistance. 

Foster  and  Porter  —  Electricity  and  Magnetism. 

Starling  —  Electricity  and  Magnetism. 

Brooks  and  Poyser  —  Magnetism  and  Electricity. 

Thompson,  S.  P.  — Elementary  Lessons  in  Electricity  and  Magnetism. 

Gray  —  Absolute  Measurements  in  Electricity  and  Magnetism. 

Palmer — Theory  of  Measurements. 


INDEX 


Absolute  measurement,  361 

of  current,  361 

of  resistance,  363 

Absorption  (see  also  Electrification), 
166 

measurement  of,  226 
Accumulator,  101,  107 
Aging  of  magnets,  316 
Air  gap  methods  of  magnetic  test- 
ing,   342 
Alloys, 

of  magnetic  materials,  314 

of  non-magnetic  materials,  315,  for 

thermocouples,  108 
Alternating  current  galvanometer,  29 
Ammeter,  18,  134 
Ampere,  141 
Ampere,  8,  9,  158 
Ampere-turn,  276 
Analysis  of  experiment,  3 
Anderson's  method,   self-inductance, 

245 

Aperiodic,  23,  209 
Astatic  system,  27 
Average  value  of  E.  M.  F.,  357 
Ayrton  shunt,  92,  95 

B-H  curves,  301,  304 
Ballistic  galvanometer,  18,  205 

suspended  coil  type,  206 

suspended  needle  type,  206 
Bar  and  yoke,  330 
Battery  cell,  101,  115 

dry,  103 

Edison-Lalande,  104 

gravity,  101 

Leclanche,  102 

secondary,  106 

standard       (cadmium,       Carhart- 

Clark,  Clark,  Weston),  105 
Battery  resistance,  78,  115 
Biot,  141 
Bismuth  in  magnetic  field,  344 


Bosanquet,  275 
Box  bridge,  72 
Burrows,  336 

Cadmium  cell,  105 
Calibration  of  ammeter,  134 

ballistic  galvanometer,  210,  217 

current  galvanometer,  44,  121 

electrodynamometer,  153 

voltmeter,  137 
Capacity,  electrostatic,  9,  15,  161 

parallel   and   series   combinations, 
169 

standards,  165 

units,  9,  162 
Carey    Foster,    mutual    inductance 

method,  257 

Carhart-Clark  standard  cell,  105 
Cell,  see  Battery,  Standard,  Storage 
Centimeter,  unit  of  inductance,  188 
Charge,  electric  (see  also  Quantity), 

9,  12,  154,  157,  163,  171,  201 
Charge,  loss  of,  95,  166,  225 
Chicago,  electrical  congress,  7 
Clark  standard  cell,  104 
Coercive  force,  306 

in  permanent  magnets,  318 
Coercivity,  281,  307 
Compensated  permeameter,  336 
Condenser  (see  also  Capacity),  161 

discharge  through  high  resistance, 

234 
Conductivity,   58 

of  electrolytes,  98 
Control  in  galvanometer,  22 
Copper  voltameter,  158 
Core-loss,  302 

measurement  of,  354 
Coulomb,  8,  9 
Current  balance,  362 
Current  inductor,  185 
Current  sensibility  of  galvanometer, 
41 


369 


370 


INDEX 


Current  strength,  5,  14,  139 
absolute  measure  of,  362 
unit  of,  5,  8,  9,  158 

Damping,  23 

factor,  209 

in  ballistic  galvanometer,  207 

in  suspended  coil  galvanometer,  28 

ratio,  209 

D'Arsonval  galvanometer,  28 
Declination,  285 
Defining  equations,  10 
Demagnetization  methods,  279,  281, 

318 
Demagnetizing  effect  of  poles,  279 

effect  in  short  bar  magnets,  281 

factors,  281 
Derived  units,  7 
Diamagnetic  substances,  265 
Dielectric  constant,  168 

strength,  168 
Dimensional  equations,  11 

formulas,  10 

Dimensions,  theory  of,  10 
Dip,  285 

circle,  287 
Discharge  key,  215 
Double  bridge,  82 
Dry  cell,  103 

Earth  inductor,  298 
Earth's  magnetic  field,  285 

horizontal  component,  288 

specifications  of,  285 

vertical  component,  286 
Earth's  magnetism,  285 
Edison-Lalande  cell,  104 
Edison  storage  cell,  107 
Einthoven  galvanometer,  30 
Effective  value  of  E.  M.  F.,  357 
Electric  current,  139 

electrolytic  effect,  157 

heating  effect,  154 

magnetic  effect,  140 
Electrification  (see  also  Absorption), 

94,  236 

Electrochemical  equivalent,  158 
Electrodynamometer,  151,  153 
Electrolytic  effect  of  a  current,  140, 

157 

Electrolytic  resistance,  96,  98 
Electromagnetic  system  of  units,  12, 
14 


Electrometer,  361 

Electromotive  force  (see  also  Poten- 
tial), 6,  8,  101,  111 
Electrostatic  system  of  units,  12 
Energy  in  capacity  circuit,  174 

in  inductive  circuit,  197 
Epstein  core-loss  equipment,  355 
Error,  treatment  of,  1 
Errors  of  observation,  2 

instrumental,  2 

systematic,  2 

E  wing's  double  bar  and  yoke  equip- 
ment, 332 

Fall  of  potential,  7,  14,  101,  140 

Farad,  9,  164 

Faraday,  180 

Faraday's  equation,  180 

Faraday's  law,  157 

Ferromagnetic  substances,  266,  314 

Figure  of  merit,  41,  42 

Fischer-Hinnen  permeameter,  339 

Flow  calorimeter,  155 

Flux  density,  273 

Fluxmeter,  329 

Flux  turns,  179 

Form  factor,  357 

Fundamental  units,  8,  10 

Galvanometer,    alternating    current, 
29, 

ballistic,  18,  205 

current,  18,  49 

Einthoven,  30 

high  frequency,  29 

potential,  45 

resistance,  76 

sensibility,  41      ^ 

specifications,  30 

suspended  coil,  19,  27 

suspended  needle,  19,  27 

suspension,  22 

vibration,  29,  248 
Galvanometers,  classification  of,  18 

types  of,  27 
Gauss,  9,  273 
Gilbert,  276 
Gravity  cell,  101 

Half  deflection  methods,  76 
Helmholtz  equation,  191 
Henry,  9,  183,  187 


INDEX 


371 


High  frequency  galvanometer,  29 
permeability,  305 

High    resistance    measurement,    89, 
235 

Hopkinson,  331 

Hopkinson  yoke,  331 

Hysteresis,  302,  307 

Hysteresis,  measurement  of, 
fixed  point  method,  348 
step-by-step  method,  344 

Hysteretic  constant,  312 

Imperfect  magnetic  circuit,  269 
Inclination,  285 
^Inductance,  9,  15,  178,  198 

mutual,  182,  183,  238 

self,  186,  187,  238 

units  of,  9,  187 
Inductance  density,  273 
-Induction,  electromagnetic,  178 
•'-"Inductive  circuit, 

charge  induced  in,  201 

current  and  energy  relations,  190 
Intensity  of  earth's  field,  285 
Intensity  of  magnetization,  271 
Interlinkings,  178 
International     Electrical     Congress, 

Chicago,  7 

International  units,  8,  158 
Intrinsic  energy  equation,  197 

Joule,  8,  9 
Joule's  law,  54,  55 

Kelvin  bridge,  79,  82 

galvanometer,  27 
Keys,  16,  215 
Kirchhoff  bridge,  79 
Koepsel  permeameter,  342 
Kohlrausch  bridge,  99 

.     Laboratory  methods, 

report,  3 

Lamp  and  scale,  25 
Laplace,  141 

Leakage,  electric,  95,  166,  225 
Leakage,  magnetic,  269,  282 
Leclanche  cell,  102 
Leeds  and  Northrup  potentiometer, 

131 

Legal  units,  8 
Lenz's  law,  23,  180 
Line  integral  of  magnetic  force,  275 


Line  of  force,  272 
Lines  of  induction,  273 
Linkings,  178,  179 
Logarithmic  decrement,  209 
London  conference,  8,  361 
Lorenz  method  for  measuring  resist- 
ance, 363 
Low  resistance  measurement,  79 

Magnetic  axis  of  the  earth,  285 
Magnetic  circuit,  264 

imperfect,  269 

law  of  the,  274 

perfect,  269 

Magnetic  effect  of  current,  140 
Magnetic  elements,  284 
Magnetic  field,  about  a  current,  141 

about  a  straight  wire,  142 

at  center  of  circular  coil,  144 

at  center  of  solenoid,  148 

on  axis  of  a  circular  coil,  143 

reactions,  5 

specifications,^ 

strength,  5,  270 
Magnetic  flux,  9,  14,  272 

leakage,  269,  282 

materials,  266,  314 

moment,  271 

polarity,  269,  270 

pole- strength,  5,  13,  14 

susceptibility,  271 

testing,  301 
Magnetism,  264 
Magnetization  curves,  301 
Magneto-inductor,  218 
Magnetomotive  force,  274 
Manganin,  61 
Maxwell,  238,  239,  275 
Maxwell's  method, 

mutual  inductances,  260 

self-inductance,  239 
Maxwell,   unit  of  magnetic  flux,   9, 

272,  273 

Mechanical  equivalent  of  heat,  154 
Megohm,  56 
Meter  bridge,  68 
Mho,  58 
Microfarad,  164 
Microhenry,  187 
Microhm,  56 
Mil-foot,  57 
Millihenry,  187 
Mirror  and  scale,  24 


372 


INDEX 


Mixtures,  method  of,  232 
Multiplier,  47 
Multiplying  factor,  33 
Mutual  inductance,  182 

definitions,  183 

standards,  183 

units,  183 
Mutual  inductance 

of  coaxial  solenoids,  184 

of  symmetric  circuits,  186 

Normal  induction  curve,  310,  325 
Normal  induction  density,  325 
North  magnetic  pole,  285 

Oersted,  141 

Oersted,  unit  of  reluctance,  277 

Ohm,  8,  55 

Ohm,  53 

Ohm's  law,  6,  53,  140 

Onnes,  59 

Paramagnetic,  265 
Paris  Convention,  9 
Perfect  magnetic  circuit,  269 
Permanent  magnets,  316 

methods  of  testing,  318 
Permeability,  189,  273,  277,  302 

at  high  frequency,  305 

effect  on  inductance,  185,  189 
Permeance,  277 
Polar  demagnetization,  279,  281 

field,  279 

Polarization,  102,  119,  121 
Poles  of  magnet,  5,  269 
Post-office  box  bridge,  72 
Potential  difference  or  drop,   7,   14, 
101,  111,  140 

galvanometer,  18,  45 
Potentiometer,  simple  circuit,  123 

type  K  circuit,  131 

Wolff  circuit,  128 
Power,  electric  unit  of,  see  Watt 
Precision  of  observations,  1 
Prefixes,  16 
Primary  batteries,  101 

Quadrant,  unit  of  inductance,  188 
Quantity  of  electricity,  9, 14, 154,  157, 
163,  171,  201,  204 

Radial  field  in  galvanometer,  29 
Rayleigh,  238 
Reciprocal  ohm,  58 


Reflecting  galvanometer,  25 

Reluctance,  275,  276,  277 

Relur  iivity,  277 

Rer  anence,  307 

"°     .dual  charge,  167,  226 

..esidual  magnetism,  302,  306,  315 

Resistance,  6,  8,  15,  53 

absolute  measurement  of,  363 

absolute  unit  of,  8,  55 

box,  62 

by  loss  of  charge,  235 

materials,  60 

of  electrolytes,  96 

series  and  parallel  combinations,  7 

standards  of,  64 

temperature  coefficient  of,  58 
Resistivity,  56 

mass  units  of,  57 

volume  units  of,  57 
Retentivity,  307 
Rheostat,  64 
Ring-ballistic  method,  322 

Savart,  141 
Secohm,  239 
Secohmmeter,  263 
Secondary  batteries,  101,  106 
Seebeck,  108 
Self-inductance,  186 

definitions,  187 

of  a  solenoid,  189 

standards  of,  187 

units  of,  9,  187 

Sensibility  of  galvanometers,  41 
Shunt,  Ayrton  universal,  37 
Shunt  box,  35 
Shunt,  constant  current,  36 

interchangeable,  50 
Shunts,  theory  of,  32 
Silver  voltameter,  158 
Slide  wire  bridge,  68 
Soakage,  167 

Solenoid,  magnetic  field  in,  148 
South  magnetic  pole,  285 
Specific  inductive  capacity,  168 
Specific  resistance,  56 
Standard  cell,  104 

Carhart-Clark,  105 

Clark,  104 

Weston,  105 

Standard  cell  specifications,  105 
Standard     cell,   temperature     coeffi- 
cient of,  104,  105 


INDEX 


373 


Storage  cells,  107 

String  galvanometer,  30 

Subpermanent  magnetism,  316 

Susceptibility,  271 

Suspended    coil    galvanometer,     19, 

28 
Suspended  needle  galvanometer,  19, 

27 
Switches,  16 

Tangent  galvanometer,  28 
single  coil,  145 
double  coil,  147 
Telescope  and  scale,  25 
Temperature  coefficient  of  resistance, 

58,59 
Temperature  coefficient  of  standard 

cell,  104 

Terminal  potential  difference,  111 
Thermocouple,  109 
Thermoelectricity,  108 
Thermoelectromotive  force,  108 
Thomson,  Sir  William  (Lord  Kelvin), 

27 
Thomson  bridge,  79,  82 

galvanometer,  27 
Time    constant   in   capacity   circuit, 

173,  174 
Time  constant  in  inductive  circuit, 

195,  196 


Torque,  19 

in  electrodynamometer,  152 

in  suspended  coil  galvanometer,  21 

in  suspended  needle  galvanometer, 

20 

Traction  method  of  magnetic  test- 
ing, 338 

Units,  7,  8 

dimensions  of,  10 
Universal  shunt,  37 

Vibration  galvanometer,  29 
in  capacity  measurements,  231 
in  inductance  measurements,  250 

Volt,  8,  9 

Voltameter,  158 
copper,  158 
silver,  158 

Volt  box,  135 

Voltmeter,  18,  45,  136,  153 
multiplier,  47 

Watt,  8,  9 
Wattmeter,  153 
Weston  cell,  105 
Wheatstone  bridge,  67 
Wolff  potentiometer,  127 
Work  (see  also  Energy,  and  Joule), 
8,  9,  154,  174,  197 


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